You can have someone grab your ankles to reduce amount of weight on the isolated muscles. How much do you weigh?
you probably just aggrevated it, and it's hard to tell how long it'll take to heal. I ripped my shoulder one time, and quit lifting completely for a few months. I came back, and could manage to do pushups, and it didn't hurt, BUT I'd get on a bench with light weight, and each rep I'd feel the muscle pulling. Took about another 3-5 months before benching, but I could do a lot of other stuff. There's tons of exercises you can do for your triceps besides dips...Though dips are one of the best. Close grip bench press, pushups with your hands close in, french curls, Dumbbell french curls (one hand at a time), DB kickbacks, skullcrushers, etc. That doesn't even include if you have a nice cable machine, or the machines to do your triceps.
The front of my shoulders. Usually goes away but it hurts considerably and i never am 100 percent. I hurt them benching.
Also might be worth checking whether the Dips bar is too wide for your frame. If it's too wide you'll put considerable strain through your shoulder.
just an update... i took your advice and went on the narrower bars (the gym has 2 sets) plus i leaned forward more and crossed my legs behind me no shoulder pain at all and all strain was put directly on my chest and triceps, i was very happy managed to do 5,5,4 at the end of my chest workout and was reasonably sore for 2 days hoping to be doing sets of 10 in a few weeks