Well, I understand this to be the main point of the question:
"the 1, 4 and 5 chords, plus their relative minors, are the only chords which are possible in any given key signature"

Well yes - that's one reason why they are the popular "3-chord trick" chords, because they don't particularly suggest you are moving off into another key.

If you want go beyond 1 - 4- 5 and relative minor (6) there are still plenty of chords, but you have to use notes that aren't in the scale of your home key, and therefore aren't in the key signature. When writing music you put sharp and flat symbols in front to indicate this. If you put chord symbols under the music or lyrics those accidentals are implied, so if something is in C and you have a D chord in it, that D chord will have F# in it, even though that's not part of the home key.

If that seems strange, bear in mind that the home key of a song or tune is only the key it starts and finishes in (usually). On the journey from one end to the other, it may well wander off into other keys, even if only very temporarily. Apart form accidental appearing in the chords, even the melody of a tune doesn't always stick to the seven notes of the scale, and then the 3 chord trick really won't do.

Anahata