Upgrading to new components should be easy enough. As mentioned above, you should go all the way rather than some kind of kluge.
You will likely need some kind of an external bottom bracket, but there are many available.
I run 9-speed in my frame at 126mm (which means 10 speed should also work). I don't know about 11 speed. Tolerances are tight. I have a mighty small outer rear sprocket (fit tight), and an off-center rear rim.
As far as spacing the frame on a small frame, the distance from the dropout to the brake bridge will be approximately the same on all 700c frames. Likewise, the chainstay length will be approximately the same, and is more dependent on the frame design. Is the brake bridge narrower than on a larger frame? It may not make a big difference. Most people clamp the brake bridge during the respacing process.
You might be able to squeeze a 10 speed hub in your space (with the off-center/asymmetric rim). It will be very tight, and may depend a bit on the right rear dropout design.
You can mount an 11-34 cassette onto a 10-speed freehub. For smaller cassettes, it will require either modifying the inside of the cassette, or buying one that has been modified. Another option would be to use a 11-speed cassette, and drop 1 or 2 sprockets (effectively changing it to a 9 or 10 speed cassette). I don't know if a 11 to 9 will fit on a 7 speed freehub, but it will be close. Perhaps I could run a test. Maybe with the 34T inner sprocket.
Picking out a stable way to modify your cassette can be tricky. If you do choose to use a 7 speed freehub, then make sure you have the 11T groove in the end if you plan on using a 11T sprocket.