Alright... inane detail work time.
1) How far is Shirou's house from the school? I need to figure out Fuyuki City distances, because I'm getting them very, very wrong.
Hmm, I'm not sure it is ever specifically stated, but it's a reasonable distance (half an hour walk or so). His house is also
not on the route to school from Sakura or Rin's house.
2) What day does Bazett summon Lancer, and what exactly happens between her and Kirei when he betrays her? Does she have feelings for him or something?
I don't think it's ever said when Lancer is summoned, but it's very early, most likely before the prologue.
As for her betrayal, Kotomine befriended her prior to the war, and she came to trust him. Then, after she summoned Lancer, he attacked her when she wasn't expecting it and cut her arm off. And, yes, she very definitely
does have feelings for Lancer. She is absolutely besotted with him.
3) Post-HF good, what can Sakura actually do? Is the Greater Grail still around, or did the Excaliblast take care of it? Not sure if he destroyed the Lesser or Greater Grail there.
It's not entirely clear what Sakura is capable of post-HF. However, she would still have excess prana, and she'd probably be capable of using most of the things she is capable of in her Dark form with sufficient practice. It would be difficult, though.
As for the Great Grail, it's destroyed. The Excaliblast destroys it and the cavern it is in.
Also, do you mean HF True or HF Normal? There's no such thing as "HF Good", and I'm never 100% sure which people are talking about, since HF True is a much happier ending than HF Normal is. I assume you mean HF Normal, though, since you're talking about the Grail being Excaliblasted.
4) When we say that Sakura has like a trillion units of prana, does that mean that it's all stored in her body? Or is it more like she can access it, but the prana's still in the Lesser Grail? Is it something else entirely?
It means she can access it, not that she has it stored inside her. Her body, like Rin's, can only store and use 1000 units at a time.
I believe that, as a former Grail, Sakura has a connection to Akasha, and thus infinite prana. She can't use it to access the source (at least normally), but she can draw prana from it.
5) What has Nasu told us about the nature of souls?
Can you clarify what you're asking here? It's too vague to answer at the moment.