When I originally got the BigRedBass, the hardware was in about the same shape. So I removed the bridge, took it apart completely and polished/cleaned up everything with brass polish, lubed the saddle screw threads, the whole nine yards. Put it back together . . . backwards! The saddles were facing the wrong way. Geez . . . So if YOU do this, I always urge people to make a little diagram first: That way you only have to reassemble it once!
Not knowing what the brands are where you're at, I'll make some 'generic' suggestions.
I use a teflon-based lube for the moving parts (saddle screws, bridge screws, etc.). The type I use is actually for firearms, a little goes a long way.
I'm sure there's several brass polishes where you are. Another thing that works is car wax, preferably carnauba-based. Takes most gunk right off, big shine, and of course leaves a film that helps with any new corrosion. Car wax is also great for getting those fingerprints off the tuning keys that don't seem to come off any other way, BUT can sometimes try to take the gold plating with it, so be careful. I'd try an out of the way, tiny spot first.
The lacquer on the bridge is a sanding job, followed by polish. I'd think there must be some sort of dip in lacquer thinner, etc., that would take it off, but I don't know what you would use. Just use a very fine grit sandpaper, and you'll end up with a satin finish, just like it's supposed to look. My Emperor Ming tailpiece is still clean, so I've managed to avoid this, so far.
I've never tried to clean a sustain block, so I'm no help there.
J o e y