I'm not sure what to tell you about your back breaking out when you sweat- y'know, when I re-started exercising after I recovered from my back injury, I remember my face broke out a bit, but after that initial period of time it cleared right up. Maybe just give it some time, and I suppose you could also use acne-cleanser made for the body (although I don't see why you couldn't just use a face wash made for acne treatment on your back)
If you want to gain weight in muscle, you are going to have to engage in some type of exercise, plain and simple. And muscle gain takes A LOT of time, especially since you are female. Thems the breaks, chica.
Working out any time is fine- just so long as you do it! Not everyone can do a 5AM workout, and that's just fine! You need to figure out what you want though- do you want visible or large/bulky (for a lady)/visible muscle mass? (Body building/ weight training) do you want lean muscle gain? (Pilates is amazing! + cardio) Do you want lean gains + endurance abilities? (Track/running/cycling). These things all come with dietary requirements too (you can't let yourself gorge on junk food as a "reward", basically- good luck trying to out-train a bad diet!)- and if you've got fat to lose, well, you're gonna have to watch your calories.
If you just want what people consider general "toning", to be a bit stronger and a little less jiggly, well, I highly suggest finding something you find fun! This could be dancing, biking, whatever gets that heart rate up and your butt off that couch! If it's something you like to do, it won't be such a chore. Much luck to you on your fitness journey!