Hi Ben, you've made a good start there, so you should be in good shape for your half in March.
For question (1) simple way to find out if its food -induced is don't eat for the three hours before a run and see if that helps, but another potential cause is running too fast. Try running slower to see if that helps, although it sounds counter-intuitive one of the ways to run faster is to do more runs at a slower pace.
For question (2) the general rule is don't add more than 10% distance per week. As you did 16 miles last week, its probably easier to add the additional 1.6 miles onto one of your existing runs, or add 0.8 miles to two of your runs. After a few weeks you mileage will increase to a level where you could spread the total distance over four runs, so it might be easier to add a fourth run and reduce the distance in a couple of the other runs to make sure you dont exceed the 10% guide, e,g. if you've run 3 x 7mile runs (total of 21 miles), you could then do 3x 6 miles plus 1x 5 m the next week.
It would also be beneficial to have one run thats longer & slower than your other runs (commonly referred to as the LSR or Long Steady Run) to get you used to running for a longer time. For a half marathon, your longest preparatory run should be at least 10 miles. Personally, I wouldn't recommend running 13.1 miles in training mainly so it remains a personal longest-distance achievement for race day.
As its your first half marathon, and you participate in other sports, you don't need to go beyond four runs a week (and three runs would probably be OK).
