Here is the setup.
B18B1 with aftermarket 2.0L Sleeves, Will be bored out due to a nick in he cylinder wall. Not sure how far yet.
B16 Vtec head.

What head gasket do I need to use? I have the 84mm head gasket already. Will that still be ok if I have to go up a size in pistons?

Full Setup:
96 Integra SE
B18B1 W/2.0 Sleeves
B16 Vtec Head W/ BC Stage 2 Turbo Cams
Titanium Valves, Springs, and Retainers
Eagle Rods, Manley Pistons
ARP Studs
Precision 5558 Turbo

 

I had to bore the cylinders to 85mm. What's the best head gasket to use and does it have to be a 85mm gasket or will the 84mm work?

 

Golden Eagle sells custom Headgaskets of different bore sizes.
I think to be safe, you would want to use the correct bore size gasket.
You may could get away with the smaller headgasket, and I'm sure it's been done before, but why risk it over 100 bucks.
I'd also call Cometic and get their opinions.
Good luck.

 

I know cosmetic sells different thickness gaskets also. What is common for a turbo setup? I already have 9.0:1 pistons and not looking for any less.

 

Stock thickness is supposed to be around .030" I believe so assuming minimal material was taken off the head and block, and the piston to deck height is about where it should be, I suppose a stock thickness HG would work.
Kinda hard to say without knowing the specifics of the build to calculate exactly what you need.