What's the consensus on just removing that board? BTW, temps are up into the 80s for now.
It depends on how much they are fanning and how congested the entrance seems. Some fanning is not an indication of anything wrong, just bees working the A/C or removing moist air from curing honey/syrup. If the entrance seems congested to the point that the fanners keep getting jostled from their positions, there are a lot of fanners (like, say, more than 5), or there is a crowd of bees waiting to enter, then I'd remove it. If you do, absolutely remove the feed (which honestly, you are probably safe to do at this point anyway). We are grazing the 80s now and I pulled the reducer from my largest colony this week.
I'll be prepared to add another deep brood box if they look like they're ready. Should I pyramid it? They seem to be doing well enough, I can't even open the side port to look in any more because it's loaded with bees in there and they come out the hole when I open it up, making it hard to put back in without killing some.
I have never done the "pyramid scheme"
(sorry, couldn't resist), so I can't speak to that. It depends on how full of bees the hive seems. Just based on how they looked last time and the growth curve they are likely on, I'm not even sure they are going to need another box, but you won't know how they really look in there until you open them up.
That box is medium and I do want a deep there. Thoughts on what to do with that? And when should I inspect it?
I also don't have experience with swarm traps, but I might be inclined to give them about a week without an inspection, just to allow them to get settled, but if all is well they should have brood in 4-5 days and by that time they won't abscond on you, so that would probably be a good minimum window. But that's just what I'd do having never actually done it before.
As far as switching out the box, you could either just trade it out if they aren't really in the medium, or if they are, you could add a deep underneath it when they need more room. By that time, I'd assume your season would be far enough advanced that if that breaks up the brood nest, it would be okay.