Is it possible to get the Companion points without having the actual companion with you? My followers are bugged because of the most recent patch, and I'd like to continue the story.

But if the companion HAS to be there for certain quests, then I'll never be able to complete their sidequests.

Does anyone know if they have to be with you in the certain situations?

This might not be what you are asking, but I have had ED-E "wait" and gotten quest updates with him parked in my safehouse. You also get the companion perks while they "wait", the only problem is you have to visit them and cycle the wait/follow command so they don't just go back to where their respective homes after awhile.

Did you send him back to the safehouse, or just go into the area and tell him to wait?

The problem I'm having now is NO other companions can join me (the join me text no longer appears after the patch) and I can only keep Boone or dismiss him.

I basically want to keep going with the main quest, but without Arcade joining me I'm afraid I'll miss out on his sidequest.

Ouch, gotcha. I just had them wait, though I traveled to the safehouse and had them wait there so I was sure they were safe. If you want Arcade though, you'll have to dismiss Arcade anyhow (one Humanoid and one non-humanoid only).