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I grabbed this one because it's the only one on amazon I could find with wide neutral grip. It's gonna come in today and I'll test it out, but only after wrapping a lot of the contact points in towels to maintain the door frame. I've had similar ones before with the same stabilizing mechanism, they do what you ask (easy up, easy down) but I've had some issues with the door frame getting lightly marked up.
Abs? Every day. Your core is working constantly to stabilize you as you go through your day to day activities, adding in some extra core work every day is fine.
Chin ups, I'm less certain on, but I'd think that you'd want to at least have a rest day following each day you're doing chin ups, especially as you add volume. Someone with more experience feel free to comment on that. I've got this pull up bar and it works just fine, no damage and I'm about 180.
Simplest solution is to buy one of those chin-up bars that fit in your door frame. I have one and use it aaaaaall the time, it's amazing.
This isn't a turks link I'm not trying to sell to you. This is the bar I use. 20 bucks, completely collapsable with the turn of a few screws and easy to assemble. Remember if you go with the bar that using a chair to support yourself through the moves is a good way to get adjusted to it, and if it's in a door frame and you're tall you'll need to bend your legs :D
If this doesn't work bands are an awesome starting point. Good luck!
Jump rope is an excellent idea, it's a great form of cardio that you will probably find much harder at first than you're expecting! Kettlebell could also be good. An adjustable dumbbell set could also be handy. Here are some other ideas:
I would recommend you check out /r/bodyweightfitness for their recommended routine or various progressions. There's a lot you can do at home!
FWIW, [link] is on sale at Amazon today for $29.