Hey Andy, welcome aboard!
Puttering? Kinda like popping through the exhaust when you let up to decelerate? More than likely this is unburned fuel getting into the exhaust, where it ignites and 'pops'. Usually indicates a rich condition, but... most bikes will do this even when jetted properly, especially under deceleration. Check your plugs, they should be a light tanish color. If they are blackish or dark brown, then you are running to rich. Something else to consider is that old plugs can also lead to false 'rich' readings, especially if the gap has worn out of spec, which will lead to incomplete combustion and unburnt fuel entering the exhaust.
No clue on the brake lines, let me know!
Yeah, the Bandit 400 has a fairly 'mushy' suspension. There are several FAQs on the site detailing how to go about changing to a GSXR rear-shock (much better) as well as a complete GSXR front end. I've got a '92 GSXR front that I plan installing on mine, should stiffen things up quite a bit!