Many pictures of the Soyuz-2 show the inside of the nozzles painted with a brilliant red. Is there any function of the paint during ignition and operation of the rocket engines, or is it just a traditional decoration? The paint will burn away very quickly.
They aren't painted red. If you zoom in on the picture, you'll see that the red parts are actually protective inserts clipped onto the nozzles to keep debris out of them during transport.
It's a widespread aerospace-industry convention to make such "remove before flight" items bright red so it's obvious if you've left one on.
Apparently the covers stay in place until very late in the launch process. In a description of the lead-up to a Soyuz launch, "Soyuz: A Universal Spacecraft" says:
The crew start the onboard tape recorder at T-15 min[utes], and ensure that their safety harnesses are tightened as the automated flight sequencer takes over. Combustion chamber covers on the first and second stages are blown off with nitrogen, while topping off of the liquid oxygen tanks is completed...
Here they're referring to the boosters and core as first and second stages.