Wow. Sorry to hear about your health problems!
This is a matter of personal preference, I suppose, but given your intro over at MDA, I would say you are closer to 80-100 lbs. underweight, not 50. [Edit: the strength guys will actually say you are anywhere from 130-200 lbs. underweight... heh.]
The simplest things for you to do are 1. eat a lot more and 2. lift heavy things. (Bodyweight exercises are great, but they don't really count as lifting HEAVY, except for very deconditioned people. It sounds like you are fairly deconditioned, but it's still not too early for you to lift heavy.)
I am 5'9" and 155 lbs. and I find that I cannot gain weight on a pretty clean Paleo + dairy diet unless I eat 4,000+ calories per day, probably more like 4,500-5,000. If you drink goat milk liberally, it shouldn't be that hard to get to 4,000 calories. Drink a tall glass with every meal, to start.
Since you are so underweight, you may be able to gain weight initially with a smaller caloric intake. I would take advantage of that in the beginning, but ramp up calories if you find that your weight starts to stall. Eating 5,000 calories of Paleo is not easy. (Drinking lots of milk can help.)
Lifting heavy things
One of the best resources for lifting heavy things is Starting Strength. It's worth many times its price ($30) and will serve you for many years. I've read it twice, reading it a third time now, and I'm continually learning from it. You can learn a little bit about the program by reading the SS wiki, but really, please spend the $30. It would be very silly not to, especially for someone who needs to gain 50+ lbs. Reading the book will save you hundreds of dollars in medical treatment for injuries later down the line.
If I were you, I would start SS immediately with very conservative weights. The idea is to add weight to the bar every time, so even if you start with a 45-lb. back squat (as I did when recovering from knee injuries), you will be squatting 200+ within a matter of weeks.
Another resource for you re: lifting heavy things is the StrongLifts program. I don't do it myself, but it's a fine program. Whatever you decide to do, it's worth reading the SL program to get additional info.
Depending on your age, height, weight, goals, and personal tolerance for goat milk, you may want to drink a ton of goat milk. Rippetoe (author of SS) recommends a gallon of milk a day (GOMAD) for new trainees who want to get stronger and gain lots of weight. Personally, I can't tolerate more than 3 glasses or so per day, even of raw milk. But if you lift heavy things 3x per week, limit your sprints, and drink a lot of goat milk, you should gain weight pretty fast. [Edit: this may not necessarily be true in your case, since you may not be fully recovered from your GI issues. It may be more prudent to adopt a SS-lite approach -- maybe lift 2x/week instead of 3x. Be patient, and be happy with slow and steady progress.]
Limit your bodyweight/sprint routines to 1x or 2x per week, and keep them SHORT, say, <10 mins or even better, <5 mins. You can get a lot out of Tabata by going for 4-6 intervals instead of 8.
Do whatever you have to, in order to get 8+ hours of sleep, and see if that helps, too.
At this point, eating a lot is probably more important than lifting heavy things, but the lifting heavy things will make you stronger, gain more muscle, and improve your health.
If, for some reason, you are really averse to doing SS, then you may be able to gain some weight by doing weighted squats, pull-ups, weighted or clapping push-ups, etc., and eating a lot. But if you really want to gain 50+ lbs., SS + lots of calories + dairy is one of the fastest -- and best -- ways to get there.
Whatever you do, pick a program and try it out for 12 weeks minimum, preferably more like 24 weeks minimum, and then make adjustments based on your results and your goals.
Best of luck!