You don't have to remove sights from the bbl before browning (or bluing), you can do it any way you want to.
Leaving them in place will sometimes give a build up of rust around those small parts and can add some 'age' to the assembly. It's just a result of not being able to more completely card those areas.
The browning soln can seep in under the sight dovetails and can rust them into place, one thing to keep in mind.
You shouldn't be applying it in such a heavy fashion that that is a problem, but sometimes it's bound to happen I'd guess.
Degrease the surfaces/dovetails & sights if needed before their last install if you are going to leave them in place for browning.
A tiny bit of oil seeping out from underneath one during the process can spoil an otherwise perfect job.
I always remove and disassemble everything to be browned or blued,,just the way I do things.
Not necessarily the only way to do things.
The tang/breech plug gets browned completely (except the threads) as it is disassembled from the bbl. Again, just the way I do things.
Pouring boiling water over the completed browning job usually gets some color change but in my experience if any at all, it's been to a slightly darker shade of brown. Not a complete turn to a blue/black color as in rust bluing.