Middle son gave me a free one last month...it had been in storage about 5-6 years.
Word on the web is that they're notoriously cold-blooded and testy to start when everything's perfect.
Choke/starter system seems to be part of the issue, as is OEM jetting. Fix is a new Mikuni VM carb, supposedly.
Mine is like yours but she'd sat for about 6 years without draining the carb prior to storage...covered with green crap inside everything. I cleaned the carb completely with no success. My fix will be to replace the carb.
Other issues may be a stuck or bent valve, per web suggestions. I ran a leak down test on mine about 3 hours ago, zero leakage from valves or rings.
This website may be the ARF of TTR's...Thumper Talk--TTR Section...
http://www.thumpertalk.com/forum/24-ttr/
In one of my past lives, I used to wrench professionally on Suzuki's, Norton's, and Triumph's...this little biatch has me somewhat stumped at the moment.
HTH, and please PM me or post here when you get her running. I will do the same.
Good luck.
-hanko