I'll tackle your questions:
1. What are some chords to use in c# minor if I've moved from E?
Is it a LITERAL transposition (ie, "everywhere you'd play an E (I) chord you now play c#, etc" for the whole progression of the song?
You can use the same chords because C# minor contains all the notes of E. The only difference is that C# is the new 1.
2. What chords (or progressions) can you use to move from E to c# minor -- (or ANY key to its relative minor)?
The main progression to use when going to a relative minor key is the 7-3-6 progression. Once you get to 6, it changes to be your 1.
3. What are some nice "color" or fill-in chords in E and c# minor?
The same chords you would use in any other key you can use for this key as well.
4. What is it about going from the major key to the relative minor that changes the sound? What does it do to the sound -- does it give the music more of a 'gospel' feel or is it a 'jazzy' or a 'bluesy' sound?
In the major key, the one is major so you have that happy, floating-on-a-cloud kinda feel. When you go to the minor key, the one is minor so you have that dark, soul-stirring kinda feel.
The thing about minor keys is if you think about them in terms of it's relative major, then it isn't that difficult to play in a minor key. You can use the same chords, notes, runs, progressions, etc. but make them fit around the 6 as the tonal center instead of the 1.