As noted, pinned/bolted badges are usually easy, just take 'em off and reapply them later. Glued on ones are tricky if you're not debadging; gluing them back on doesn't always go well.
Polishing and product removal strike me as two separate issues.
Removing dried product residue:
Brushes never work for me; the only brushes I've ever used that were aggressive enough for that but still gentle enough that they didn't mar paint were from TOL...long discontinued (this was back in the early '90s) though I still have a few around somewhere. A small natural-fiber brush (e.g., BHB, china brushes) soaked in QD can work, but by the time it's soft enough that it won't mar it's usually too soft to get residue off very well. The brushes supposedly *made* for this have all marred paint for me, as will a dry BHB..under magnification the scratches were just awful. I simply don't know how people use them without marring.
If a spritz of QD and a very plush MF won't do it, I use QD-dampened cotton wool (lady's makeup balls or the tuffs from the head of Q-tip) worked with a sharpened wooden swab stick. Suede-style MFs can work well for this too (the sticks poke through regular MFs too readily for me). Gotta be very careful that the stick doesn't poke through whatever you use and mar the paint, and even just the concentrated force can be too much for some paints. On harder paints, wax/sealant residues can sometimes be removed safely with a flagged swab stick (flag it by cutting with a razor blade) soaked in QD until the wood is soft, but man-oh-man is it risky.
Polishing/applying product:
For polishing, I use the above stick-cotton/suede MF method to apply the polish and then I buff it away before it dries with a very plush MF using #34 as needed. Yeah, this can take a long, long time if I'm correcting marring on hard paint.
For LSPs, as long as I apply them nice and thin the nap of my MFs gets it all. If I'm a little heavy-handed when applying I'll use some QD, but if I really overdo it I wipe everything away immediately and reapply with the proper amount.
Heh heh, this whole topic is one more reason why I debadge most of my vehicles, at least to some extent
