No bill of attainder involved. Congress is empowered to impeach without involvement of a court. The narrow definition of “attainder” is legislature actinng without a court. By that definition, all impeachment would be bill of attainder. Impeachment is “okay” cause the penalty is limited and the penalty is not criminal.
No ex post facto violation. There is an issue about whether or not the alleged offience is impeachable, but that standard is up to Congress. What constitutes impeachable has no grounding in principle, other than purely academic arguments on both sides.
Congress is only empowered to impeach and remove a president, vice president and civil officers.
This would be a bill of attainder.
The Constitution says impeached and removed from office. If you are not in office, you can’t be removed, therefore you can’t be impeached.
Mootness is very much at issue.