I just started to learn jazz guitar last year, at the tender age of 40. I'm absolutely loving it, and it's given me a deeper understanding of jazz bass playing too, so it's a win/win situation. Once you get basic (moveable) chord shapes under your fingers then you can literally play anything. I sight-read my way through 2 jazz standards the other night - something that was unthinkable last month.
I'm on a Vintage (brand, not age) SG copy with flat wound 10s, which make bending a bit difficult due to the tension, but as a 'second' instrument I don't want to be shelling out to restring it every couple of weeks.
As for choice of guitar, I have recurring tendonitis in my fretting hand, so had to be careful about neck profile etc… I really wanted to play a Telecaster (who would't?) but the neck profiles were a bit too thin for me to comfortably play for any length of time; also the slightly shorter scale-length of Gibson-style guitars made more sense too.
34 is nothing.
Music is everything.