Hi Chris,
When you installed the new mufflers did you have a SERT or Power Commander or similar device installed? For 2006 there was a factory download to set the fuel settings if you got more open mufflers and a more open air intake. You may want to try the download if you get a different air filter such as a Ness Big Sucker.
otherwise you may want to have you bike dyno tuned to make sure the Air/Fuel ratio is rich enough. (it comes lean from the factory which makes it run hotter.) If this is your first 103 motor you will notice more heat over a smaller stock motor but the 103 is no where near the heat level of a 110 -they are very hot!
You may want to double check your heated seat controls too to make sure they didn't get accidentally turned on too.
1. the shake is normal due to the rubber engine mounts and the large displacement, but the rubber can deteriorate over time so you may want to check them out. I think the front mount is the first to go but maybe one of the site mechanics will offer their expertise soon?
2. see above about heat, in addition consider removing the lowers in warm weather. You may also want to consider true dual head pipes to get the rear pipe away from you somewhat.
3. lowering can be done with shorter sprung shocks but you may loose some travel in your suspension. The expensive fix is a set of Legend air shocks with the on board compressor. You could lower the bike when parking and raise it up on the road just by pushing a button.
4. some members swear by the stabilizers and other members say they couldn't tell a difference. I have had my 2006 since new and i try to keep up with air pressure in my tires and keeping them balanced as well as proper adjustment to the steering head bearings and propper pressure on my rear shocks. So far I have not experienced any wobble on my bike. Other have experienced it though. I would say if you have experienced any wobbles then consider the maintenance issues and if you are still experiencing the wobble or you want piece of mind then by all means get a stabilizer.
5. I have not used one but several people I ride with have tried the pads from Propad and they love them. Seems like a cheap try for comfort vs. a rebuild on your seat or a new aftermarket seat. If you go the new seat route you could consider a lower seat to get you closer to the ground vs. lowering the whole bike.
Good luck and ride safe.