ZX9R B3's had carbies. Hence, it sounds like a Jetting or needle height issue more than anything else. Has it got the standard pipe or an aftermarket one??? Other solutions could be carburettor synchronisation, Plugs or even leads. My best guess though would be carbs. Best fix is to not ride it just off idle!!!
As for suspension, start by setting the rider sag. Measure the distance, from a static point-ie;from a point on the ducktail to the axle at the rear......the front should be self explanatory....if it's not, then i'd suggest enlisting someone more knowledgeable to assist you), with the suspension completely unweighted/unloaded. Hop on the bike and then measure from the same points. Aim for around 30mm front and rear difference. From there, it's about fiddling with the damping till it's right. If it has standard suspension, then it's pretty much next to useless. I would crank the damping to max on both the front and rear. Even then, it's gonna wallow still. If you really want to fix it, replace the rear shock and have the forks resprung and revalved.
As for Tyre pressures....this is a piss take isn't it!!!!

You'll get loads of opinions and lots of people will tell you different, but the B's are a heavy unit and as such, I would run between 32-36 Psi in the front and 38-42 in the rear, depending on the brand of tyre.
Good luck with the 9!