So how does staying at a location during the change over work? Just show up in which ever area I want to stay in at 5:00pm and show that I have an HHN pass?
Do any of the areas ever get too full and not accept anyone at a certain point?
Yes, starting around 4:00, they scan your ticket, give you a wristband, and let you into the roped off holding area. (Finns will generally have 3 people scanning, you enter by the Christmas shop--yes, it gets that busy.) If you are already in the area--say, at the bar--they will walk up to you soon after 5:00, scan your ticket with a handheld device, then give you a wristband.
I've never seen one close due to capacity, but I've been in Finns during holding when the building was obviously well over capacity. The walkways remain open as well, so not sure how you would even measure the legal capacity of the entire holding area. But I don't think they're counting anyway.
As mentioned above, Diagon was a holding area in 2015, was not last year. Probably related, Leaky Cauldron was open during the event in 2015, not last year. But keep in mind everything is subject to change the first couple weeks of the event. We might get a new holding area, or a traditional area might close.
One thing about eating at Finns--every table is filled, and it's hard for staff to navigate the crowd, so expect slower than typical service. If you sit anywhere close to 5, you will likely not be done with your check paid when houses open (5:45ish), defeating the whole purpose of Stay & Scream.