Yesterday, I forgot my straps and got an opened callus on my hand during wide grip pull-ups. Some guy in the gym gave me an elastic band and I continued exercising, though weights were a bit less (mainly because of not having straps, on lat pull-down, bent over row etc). Earlier when I wasn't going to gym I didn't use straps for pull-ups and didn't have problems with calluses. Later on when I stated attending gym I found straps useful as they allowed me to use more weight. So, I guess constant use of straps softened the skin on my palms. What is a good practice with straps? Should I sometimes do sets without straps with lesser weight? Also how to treat open calluses and exercise with them? Is wrapping it up with elastic band good enough?
My personal philosophy for using straps is to use them when I don't want my grip strength to limit the effect an exercise will have on a muscle group I'm trying to work. If I'm doing Romanian deadlifts, barbell shrugs, barbell rows after deadlifts (conversely, I don't use them on Pendlay rows since I reset the bar between reps), or Kroc rows, I'm more likely to use straps.
However, using straps ultimately needs to coincide with your goals for weight training. Some people won't use straps for anything, so that their grip strength gets worked on anything and acts as a natural limiter of overreaching. On the other hand, using straps is sometimes necessary if you're always trying to lift as much as you can. They are also useful for working around small injuries to your hands (such as jammed fingers, torn calluses, etc).
When it comes to treating a torn callus, treat it like any other kind of open wound. How bad is it? If you tore fresh, healthy skin, I would use an antibiotic ointment and sterile bandage. In the past when I've torn a callus, I lightly bandaged it with a small (~1 square inch) sterile pad of gauze and used athletic tape to keep it in place. Keeping the bar at the proximal interphalangeal joints of the hand (between the first and second finger bones) will help keep the bar from damaging your calluses. Here's a video where Mark Rippetoe discusses callus management.
I'm in the no-strap no-glove camp. I trained for a while in powerlifting, and if you lift raw you can't use straps.
So I'm of the mindset that a deadlift is how much you can lift, and that includes how much you can hold onto. To that end, once I get up north of my 5RM I'll use a mixed grip. I think if you need help holding onto the bar at 5RM or lower then you probably need to work on your grip, which the deadlift (and weighted pullups) helps a lot.
I did a lot of training for a while with kettlebells, and in particular the snatch can be rough on the hands. With those, the bar is rotating in your grip which pulls on the skin, and the "cardio-esque" nature of kettlebell training means you might be getting sweaty too.
I keep my hands dry and chalked when lifting or on gymnastic rings, and have relatively minor calluses and haven't had a rip in years.