I'll describe only thoughts when play by ear, because it's mostly required:

RosinedUp said
Here are some possibilities that come to mind for me:

* Most obvious: the fingering of the note.

Yep, but only after the next question:

* What the note is going to sound like.

EVERY TIME!!

* An image of the letter name of the note---do you see a letter in your mind?

Never

* The sound of the letter name---that is, do you say the name of the note in your mind or even aloud?

Never (i have a big NO PERFECT PITCH)

* Even though you are not reading from sheet music: an image of the note's position on the staff.

Only if i want to wright it down. Which is reeeeaaaly seldom. I find the sound first - then i figure out what note is it.

* The representation of the note in other notation, such as ABC or tablature.

Nope.

* The interval between the current note and the next note.

Yes. If You mean difference of pitches - so i could figure out how far i should go to get the next note. And No - if You mean the fixed interval like 3rd..4th..5th.. octave.. etc.

* The interval between the tonic and the next note.

Definitelly - no! =)

Can you mention things that I have not thought of?

Usually when i play something by ear - i just need the first tone, then it goes by itself.. I didn't think about how to.. But i definitelly don't use any scientific background for it It needs some time for me to learn the tune by ear, but it's easier than learning from the sheet music. If i doubt in a part i should play (when there several parts sound with equal volume) - i cheat and just play what fits in LOL!

About improvising: i can't improvise, until i'm familiar with the tune. I guess right here could be really useful knowledge about scales, note identification (perfect pitch), intervals. But my brain doesn't work that way. 

Do you think of different things at different times?

Yes. Intonation, dynamics, vibrato, next note, shifting to the next note, analysing what was just played and how, what to fix. Though mostly it's a combination of several of those at the same time