To answer your last question first. You normally don't need to redish unless you are adding 2 or may 1
cogs.

I have
Shimano
- Dura Ace - very quiet, nice finish, shaped teeth
- 600 - similar, maybe not as nice a finish.
- Z012, Z15, others. Lesser finish. Sometimes noisery.

Maillard
- teeth aren't as great for shifting, but still work. Somewhat noiser.

Sachs from early 90s.
- nice finish, although it can flake off. Fairly quiet. Slight improvement on tooth design of Maillard.

Suntour.
- Various qualities. Some quieter then others. Tooth design nothing to write home about.

Sunrace. Sort of a knock off on Shimano. Seem OK, don't have have a lot of miles on them.

Regina - The corsa look pretty crude. The Americana (or is it America) looks pretty nice.

Never looked at IRD.

If noise and finish aren't a big deal for you I would go with the Sunrace. Or maybe a Shimano.
You didn't say how many cogs.