Guest List Question

For me, it would matter more whether you sent her a Save the Date. If you've already communicated that she should clear her schedule that day because an initiation is forthcoming, she should probably be invited. It might provide a great opportunity for you to reconnect!

I think she should still be invited. Anyone invited to pre-wedding events like the bridal shower or bachelorette party (regardless of whether they were able to attend) should also be invited to the wedding. She might decline if she doesn't feel like the friendship is as close as it used to be, or she might attend and you can use that as an opportunity to continue the friendship!

If you haven’t communicated recently, I would find it odd in her shoes to be invited to the bachelorette. Typically it is similar to a shower where only those invited to the wedding are invited to attend. If you don’t plan to invite her to the wedding, don’t invite her to the bachelorette, even though a year has passed.

Yes you should invite her.

Only invite her if you truly want her there. If you’re not close to her anymore and don’t plan on ever talking to her again then don’t bother. Honestly, she probably won’t even attend if you guys aren’t that close anymore. You don’t have to invite anyone you don’t want to. And judging by the post it sounds like he really don’t want to invite her, so don’t.

If you don’t want to invite her, don’t. Everyone has different criteria for who they want to be there. There will be people at mine that I haven’t seen in a decade and only talk to every few months - this is par for the course with friends and colleagues in both my and FHs lives. Plenty of people wouldn’t dream of inviting someone they don’t get together with every few weeks.
If you would only be inviting her out of obligation and hope she doesn’t come, don’t invite her.