[SDT] "And G-d spoke to Moshe and Aharon saying... DABEIR (you Moshe, not both of you, DABRU) to the children of Israel... Only Moshe could tell the people about the PARA ADUMA, which is an atonement for the Sin of the Golden Calf. Aharon was too involved in the Golden Calf episode. He didn't tell this mitzva to the people and he didn't prepare the PARA ADUMA, his son did. Yet the pasuk tells us that G-d spoke to both Moshe and Aharon. Perhaps this contains a private rebuke by G-d to Aharon... And perhaps a bit of the opposite as well.
[SDT] Rashi says that the mitzva is for the assistant Kohen Gadol to tend to the Para Aduma, although any kohen qualifies. Commentaries see a symbolism in the son of Aharon doing the work: just as the cow atones (so to speak) for her calf, so too the son atones for his father who was somewhat involved in the Golden Calf.
"Take a PARA ADUMA T'MIMA..." T'MIMA usually means blemish-free, fit for the Altar. However, here the word T'MIMA is followed by the phrase "that has no MUM (blemish)", making the adjective T'MIMA superfluous.
Therefore, we are taught that T'MIMA in this context is describing ADUMA, indicating that COMPLETE reddish hair is required. Without T'MIMA, a cow that was a "jinji" would be acceptable even if it had some non-red hairs. Not so, because of the word T'MIMA.
As opposed to all korbanot in the Mikdash which had to be brought "inside", the Red Cow is slaughtered and prepared "outside". It is not a korban, but it does have korban-like features (atonement, among others). After the cow is slaughtered, it is burnt whole (some of its blood having been sprinkled towards the Mikdash first).
The complete process of the Para Aduma (including what is thrown into the fire, how the ashes are collected and how the potion is made) is a positive mitzva  that has been fulfilled nine times, so far. The next (tenth) time will be in the time of the Moshiach.
A person who comes in contact with a dead body is rendered ritually impure for a seven-day period . The "Para Aduma Potion" is to be sprinkled on the defiled person on the third and seventh day. Without this procedure, the state of ritual impurity remains forever. It is most important to avoid entering the Mikdash while one is defiled.
Intentional violation is a (Divinely imposed) capital offense.
 A kohen must still avoid contact with a dead body (except those of his close relatives for whom he sits shiva), even though he is already "tamei". This is both for "practice" as well as not to "add" to his state of TUM'A.
(3) Some gifts of the Kohen (such as t'ruma, t'rumat maaser, challa) are not given to a kohen, but are "disposed of" according to alternate halachic procedures, because of TUM'A of both the Kohen, potential recipient, as well as the giver, and therefore, the gift itself.
Note that there are gifts to the kohen that pose no TAMEI problems; these are given today (e.g. Pidyon HaBen).
[SDT] Many forms of ritual impurity require immersion in a mikve, and that's all. Mikve is done by the person him/herself. Not so with ritual impurity due to contact with a dead body. Mikve is part of the purification process, but so is the Para Aduma Potion. And that cannot be done alone. Another person MUST be involved. Think about what it means.
Note that the cedar branch and hyssop are added to the potion as well as to the burning of the Para Aduma. Commentaries see special significance in the fact that the cedar is a lofty tree and the hyssop is a lowly shrub.
The dual nature of the Para Aduma potion (that it purifies the defiled and defiles the ritually pure) is counted as a mitzva of its own . (Some draw an analogy to some medicines that can cure an ill person but would make a healthy person sick. The Para Aduma is one of our spiritual medications.)
The next topic the Torah deals with is the death of Miriam in the Tzin Wilderness in Nissan (the 10th). The People had no water (Midrashim speak of the Well of Miriam that miraculously accompanied the People during their wanderings. This well disappeared upon Miriam's death, since it was in her merit - because she had watched over Moshe at the river - that we had the Well.) The People complain bitterly to Moshe and Aharon. (The custom of emptying out water containers in the room in which someone has died, comes from the sequence: "...and Miriam died ...and there was no water...")
G-d is "angry" at Moshe and Aharon for missing a chance to sanctify G-d's Name by having the People see water come from the rock by speaking to it. (The People had previously seen water come from a struck rock.) G-d decrees that neither Moshe nor Aharon shall lead the People into the Land of Israel.
(Because of the inclusion of Aharon in this decree, there is an implication that he was not held accountable for any involvement in the Golden Calf - a point that needed clarification. Rashi says that the Torah is telling us that Moshe and Aharon would have gone into Eretz Yisrael, except for this, and only this sin. Interesting that Moshe himself tells the people (in D'varim) that he carries some of the blame for the Sin of the Spies. With Aharon's involvement in the Calf incident and Moshe's in the Spies episode, there is an interesting balance. On the other hand, Aharon IS held accountable in this case, even though it was Moshe who "called the shots".)
G-d's decree seems excessively harsh on Moshe and Aharon. Commentators point to this as an example of how strictly G-d judges the greatest of our people.
(3) A twist on the Chilul HaShem possibility of (1) is that G-d wanted to avoid Moshe's losing face. G-d and Moshe are very much partners, so to speak, in the perception of the People. At the Sea, the people believed in "HaShem and in Moshe His servant. In contrast, their lack of faith is expressed as their talking against G-d and against Moshe. These are the only two times the word U'V'MOSHE appears in Tanach.
[SDT] In asking for passage through Edom territory, Moshe's messengers state that the people "will not drink water of a well". Rashi says that we would have expected the Torah to say "the water of cisterns". Rashi explains that Edom had the cisterns; we had a miraculous well (as well as Manna for food). What we were offering Edom was the profits from selling us food and water. We had no need for their food and drink, but it was a proper offer to make. Rashi says that when staying at an inn, one should partake of the inn's meals rather than "brown bag it". This increases the benefit to the hotelier.
Moshe sends a message to Edom saying, "...you know all the trouble we had in Egypt." Imrei Shefer asks, how was Edom expected to know what happened to us in Egypt? The answer, he says, comes from Parshat To'l'dot, when Rivka sought out G-d to explain what was happening inside her. She was told that the twins in her would grew to head great nations, and when one fell, the other would rise proportionally. Edom's life must have made a significant turn upward during the dark years we spent in Egyptian servitude. That is how Edom would know what was happening to his brother Israel.
The Midrash says that the Heavenly Clouds that protected the People, left upon Aharon's death.
[SDT] We can see now that the miracles of the Midbar were each associated with one of our leaders: Moshe, the Manna; Aharon, the Clouds; Miriam, the Well.
That made them vulnerable to attack from Emori. The attack was successfully countered by Israel. The People then tired of their extended travels and complained once again to G-d and Moshe. Their tirade included gross disrespect to G-d's miracle of the Manna. For this they were punished by an attack of "fiery" (poisonous) snakes that bit many people, causing many deaths. The People repented and pleaded with Moshe to pray to G-d to spare them. G-d told Moshe to fashion a copper (the choice of copper was Moshe's and it was a play on words) snake and mount it atop a staff, so that anyone who would see it (i.e. turn their eyes and hearts towards G-d) would live.
The Mishna in P'sachim (4:9) records different things that Chizkiyahu HaMelech did, and was either praised for them, or not. He destroyed the Copper Serpent and the Sages approved of his actions. People were misusing it, and misunderstanding it.
[SDT] It is important to note that Israel's military might is not absolute, nor are their military options equal. Israel fights against whom G-d tells us to, and we do not engage in battle anyone that G-d forbids us to. It is irrelevant whether Edom was stronger or weaker than Emori. We didn't fight the latter and avoid the former for military reasons. G-d is the One in charge. We have to always keep this in mind; and it would help if our enemies knew this as well. Ironically, it is our enemies who sometimes seem to believe in G-d's role in these kind of matters, whereas we sometimes seem to stubbornly deny His role.
[SDT] Rashi explains why G-d had to tell Moshe not to fear fighting Og. Og was the sole survivor of the Flood (except for Noach and company), and he was the one who told Avraham that nephew Lot had been taken into captivity. Perhaps he had earned enough merit to resist the Israelites. G-d told Moshe not to worry.
Israel's military victories in the Midbar, towards the end of the period of wandering, were very important for the morale of the people as they faced long years of many battles upon crossing the Jordan River into Eretz Yisrael. In the Midbar, they get a taste of G-d's promises and might.
Moshe sends Meraglim to Ya'zer. Rashi says that the spies that were sent said, "we will not do as our predecessors did; we have complete confidence in the power of Moshe's prayer. In a way, the sending of these Meraglim is a TIKUN (repair) of the Sin of the Spies. Spies were always sent to facilitate the nation's next step. They were not meant to suggest yes or no.
The final pasuk tells us that Israel traveled and arrived at Arvot Moav - this is their final stop before entry into Eretz Yisrael.
[SDT] Note the significance of the above statement. The four remaining sedras of Bamidbar and all 11 of D'varim are still in front of us, and we are already at Arvot Moav. Mas'ei will give a summary of the wandering, but with the conclusion of Chukat, we have arrived at the threshold of Eretz Yisrael. Remember that back in Mikeitz we left the Land and went down into Egypt. Now we are readying ourselves to return.
Two further points about the Haftara from A Haftara Companion by Rabbi Julian G. Jacobs.
Rabbi Jacobs points out the the Haftara, in retelling the story from the Torah of Israel's sending a request to Edom to pass through its territory, the Navi also says that Israel requested the same of Moav. The Torah does not mention that. (Perhaps there is an allusion to it.) Yet when the Torah explains why we to not accept a Moavi into K'hal HaShem, it says that they were not hospitable towards us during our wandering in the Midbar. Clearly, Moav was approached and refused us passage, i.e. hospitaliy, but the Torah does not mention what happened with Moav specifically. The Haftara fills in the details for us.
Secondly, Rabbi Jacobs offers a connection between the Yiftach story and the sedra. Both the people at the end of their wandering, and Yiftach in his time, swore to G-d that if they were victorious over Emori, then they would... dedicate the spoils of war to G-d... they would offer to G-d the first to come out of Yiftach's house. In both the sedra and haftara, a promise was made, Emori was vanquished, and the conditions of the promise were kept. A tremendous difference, however, on that last part. Whatever exactly happened to Yiftach's daughter, it was an ill-fated condition of the promise that Yiftach offered. In the sedra, the condition was quite normal and acceptable.
Rabbi Jacobs ends with the suggestion that there is a parallel between the enigmatic Para Aduma and the equally (?) enigmatic choice of Yiftach to lead the peaople. He, apparently, was not a suitable candidate for the job.