That's true about Severide, Shay was in the hospital. I really, really want a Shay/Lindsay scene or a Shay/Severide/Lindsay team.
I don't think Severide had been to Erin's apartment before the night he turned up saying he didn't know where else to go, I just assumed maybe that he had called her before hand and asked to come over and that's why she didn't seem surprised. I'm sure she'd given him her card in the previous Fire episode.
As for Burgess and Ruzek, that could be true. I'm never entirely sure though how bothered Voight is about Halstead and Lindsay not going out because they're in the same unit. I can't figure out if that's just a bit of an excuse because he's protective over Lindsay but didn't actually want to say that to her. I imagine it would bother him a bit though.