If soloing over a song in a major key (let's say B major), would it generally sound better to use the B minor pentatonic scale? Or would you want to use the relative minor pentatonic scale instead

If your song is in B major, you'd want to use B major, unless you have a really good reason to use a minor scale to solo over a major chord progression.

If using the G# minor instead of the B minor scale, I would assume you still want to emphasize the B notes in the solo, right?

Since those scales have the same notes, emphasising the B would mean that you're essentially playing the B major scale, not G# minor.

if the notes in a relative minor and the major scale are exactly the same, how could the starting point of the scale alone make one sound happy (major) and the other one sad (minor)?

They don't. It's the intervals of the scale that creates the tonality. Relative major and minor scales share the same notes, but a different set of intervals.

let's say you have a song in C major with a 1-4-5 (C-F-G) chord progression. Could you not also refer to that same song as being in the key of A minor with a 3-6-7 (C-F-G)

Kinda, but you usually wouldn't as it doesn't really help anything. Since your chord sequence doesn't include Am, there's really no point in thinking of it as that key. Now, if your chord sequence was C, F, G, Am, you could think of it that way, depending on where the sequence resolved and which chord felt like "home".