Kira, you did a great job! You figured out the sweep-range that needed to be adjusted with the limit screws.
The double-click to go from inner to middle-chainring is caused by too loose cable. You're off by one click. Here's what's happening:
1. When the shifter is in the inner-ring position and the FD & chain on the inner-ring, the 1st click does nothing but take up the slack
2. Shifter is now in middle-position, but FD and chain are still in the inner position.
3. Click a 2nd time and shifter is now in outer-position and FD/chain moves from inner to middle-position.
4. no more clicks available and you can't shift to outer chainring.
To resolve this issue:
1. place bike on stand
2. spin crank and shift FD to the outside as far as it will go
3. spin crank and move the FD by hand all the way to the outside and have chain shift to outer ring. stop.
4. push shifter all the way into inner-most (granny-ring) position, but DO NOT spin the crank. The chain on the outer-ring will keep the FD from moving all the way in. You will have a lot of excess cable slack.
5. loosen cable-clamp bolt on FD gently
6. pull through (tighten) about 4-5mm of cable
7. tighten cable-clamp bolt.
8. test FD shifting now, loosen/tighten cable-tension with barrel-adjuster to fine-tune.