I haven't seen this addressed anywhere, so I'll ask; can you assign feats to bystanders? I think you should be able to if they meet the prerequisites.

If you can, what happens if you assign Lazarus pit to one? would you be forced to return them to the field the next turn since they don't have any clicks to heal?

You sure can. Not much point in doing so, but as long as they meet thr prereqs, you can assign feats to them.