Hi thank you for your question. Please remember to RATE my answer OK SERVICE, GOOD SERVICE OR EXCELLENT SERVICE so I can get credited for my time.
I am sorry to hear about your situation, unfortunately, this is something which is occurring more and more now, where applicant's after obtaining indefinite leave to remain are ending their relationships with their partners and starting new relationships.
The fact is that after obtain indefinite leave to remain it is very difficult for the Home Office to revoke this and they only revoke indefinite leave to remain in rare circumstances, please see the following link:
As you will note is only in deception cases where fraudulent documents have been submitted or if the applicant spends more than 2 years outside of the UK that his indefinite leave to remain will be revoked. Going back to your specific questions:
1. Yes he can file for a divorce himself there is nothing preventing him from doing so.
2. Whether is effects his status will depend on if you or someone else report him stating that he has been deceptive and has lied to you throughout your relationship,only then will his indefinite leave to remain may become of interest to the home office.
He may have to explain if he decides to apply for British citizenship why and how his relationship ended with you. But provided no one reports him and he files for a divorce and initiates the proceedings then it should not effect his indefinite leave to remain status. Should you wish to report him to the home office on the basis that he has been deceptive you may do so by using the following link:
I hope this answers your question if so kindly rate my answer positively so I can get credited for my time. If however you feel that the answer does not cover all the points raised in your question please DO NOT rate my answer negatively I will be happy to answer further question until you are satisfied with my answer.