Lead Tidbit
Being TAMEI is no sin, and yet...
Very unusual - maybe the only time... Rashi's commentary on Parshat Chukat begins with about 30 Rashis spanning 14 of the 22 p'sukim of Perek 19, the chapter dealing with Para Aduma. That part's normal. But then there is another set of Rashis going over the first part of the Para Aduma portion, with different comments all based on the same theme: That the Para Aduma is an atonement for her son (so to speak), the Golden Calf. Rashi, based on the teaching of Rabbi Moshe HaDarshan, shows many analogous points between the Red Heifer and the Calf, the result being a very convincing demonstration of the notion that Para Aduma is indeed a KAPARA for Cheit HaEigel.

Except for one (perhaps) major point: The Golden Calf was a sin; the topic of Para Aduma deals with ritual purity and impurity which involves contact with a dead body. Being Tamei or becoming Tamei is not a sin. It would be a sin to enter the Mikdash in a state of Tum'a. Or to eat sacred foods. Or for a kohein, with certain exceptions. But it is not per se sinful. In fact, contact with a dead body is often a great mitzva, and an act of Chesed shel Emet.

So what's the connection between sin and atonement, on the one hand, and Tum'a and Tahara, on the other.

It can be said that G-d originally planned that people would live forever. Adam & Chava's sin brought death to the world. Modified plan, so to speak, was that the people of Israel upon receiving the Torah, would be on such a high spiritual level, that they - as a subset of all humans - would live forever.

When 40 days after Matan Torah, they sinned with the Golden Calf, the plan changed again (so to speak) and there would, once again, be death. And that brought about the existence of TUM'AT MEIT, the ritual impurity from contact with a dead body. Therefore, the sin of the Golden Calf brought about Tum'a and the Para Aduma, whose ashes rid one of that Tum'a, is an atonement for that Sin.