Can Assembly has different version of BOM?

Yes you can use older revisions in your BOM! In our company we do exactly that. But you may need to customize Innovator a bit when you want to do this on a regular basis. 

Question: Do you need this on a common basis? Or just now for this particular BOM? Are you in general fine with the behavior of the BOM revisions, or is it a constant pain?

By default the Part item selector will only pick the latest revision on a Part. But you can also pick older revisions intentionally by using the AML search or by using a custom search mode. But I am not sure if the selected previous revision will float automatically to the latest revision. Of course this action still requires you to have edit access to the item. You can temporary grant Admins the permission to edit this Part without change control, but I don´t think this one is the clean solution you are looking for on the long run.

By default Innovator uses float behavior for the relationships. The old revisions will float to a new revision automatically. In our case we changed this one to "fixed" behavior. This has a few downside effects - now you have to take care for every (!) revision change by yourself. In our case this is what we want - nothing shall float. But it doesn´t fit to every company. If you want 99% of your items to float, this might not be the solution for you. If you want to use fixed behavior like this, test it out in a test database first! It´s even possible to mix fixed and float behavior, but keep in mind that this one requires further effort to make it transparent to the end user.

BTW: From a personal POV, I always wished we one day would get a "controlled float" behavior mode in Innovator:
How can I control/track floating behavior of Parts? | Aras
I developed a custom CM process to support controlled float, but I mainly use it for secondary items like Documents. 

This answer will probably not answer all question that you will have right now, but it´s a start :)