Recommendation letter from a research adviser who is expelled from the university
This is a tricky situation, but as long as everyone is careful and honest, you should be fine.

Here's what I would recommend.  Yes, you should ask your former advisor to submit a reference letter on your behalf, from his non-academic email address.  You should mention in your statement that he has left your university, but you should not say anything about why he left.  Any further details are up to your former advisor to reveal, if he wishes to do so.

Your former advisor absolutely should not submit his letter on university letterhead, because he is not affiliated with the university.  He should include a very brief bio describing his former affiliation and listing his departure date.  If possible, he should include some external evidence of his scholarship, like a pointer to his Scopus or Google Scholar profile.  If he wants to describe the circumstances of his departure, he should keep the story very brief; his letter should focus on you, not about him, his former employer, or their political conflicts.