If you are going to keep your stock 1999 wheels, then I don't suggest using spacers over one inch thick, since thicker spacers can increase leverage to a point that may be detrimental to your wheel bearings.
Yes, bigger tires can potentially affect your transmission's performance. Bigger tires may cause your transmission to "think" you're towing a load and might no longer shift into overdrive and/or not lock up the torque converter when it should or other behavior. Changing the axle gear ratio is how you compensate for this, though regearing axles is far from an inexpensive proposition...
I would suggest drive it a bit first with the new tires and see if it behaves OK... 285/75R16 is not a terribly gigantic increase (not quite like going to 35" tires would be), so it may be fine, though that is probably the biggest size you could go and still have a chance of it working OK. Your speedometer will of course be a little bit off though (will read a few MPH slower than your actual speed)
Also, technically those are called Stabilizer Bar End-Link bushings, but many (most?) people do refer to them as sway bars, so most places (where there is a human present, anyways) should know what you're inquiring about.
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