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!
Alright, because of your money the solution for you is called "Calisthenics".
Lifting is a form of resistance training, and it uses weight as the resistance. Calisthenics uses your body-weight as the resistance, saving you money while still very effectively building muscle and improving your cardiovascular system.
Here's an introductory video demonstrating a beginner's calisthenics workout: https://www.youtube.com/watch?v=HYFDcpX9oMQ
Notice that you can replace anything requiring a resistance band with a "negative" variation of the excercise, which means you do not need to purchase a resistance band. Also, for the third exercise (The alternative dips), you do not need a lower bar like the one that she uses. Simply use a chair.
If you cannot find an outdoor gym similar to the one that you see in the video I provided above, then these are the things that you will need to purchase. I have provided an amazon link to the best for price for each:
Yes... The dip station does cost a bit, and together this is a bit of money. These purchases will give you everything that you need in order to progress through calisthenics, though. For lifting, you would either need to continue to purchase heavier weights or to purchase a gym membership. In other words, this is the most cost-efficient in the long run.
For diet, I recommend checking out this sub's information hub regarding diet. Here's a link: https://www.reddit.com/r/Fitness/wiki/getting_started#wiki_fixing_your_diet
Check out /r/bodyweightfitness for more info. If you have any questions at all, you can also respond here.
Here are is an intermediate workout once the beginner one becomes easy: https://www.youtube.com/watch?v=Q2xPSFFoJVQ
Here is an advanced workout once the intermediate one becomes easy: https://www.youtube.com/watch?v=Q-3n01R_XBA
Got this from Amazonduring 2020 lockdown. No installation and I use it pretty frequently still, both for pull ups and just to stretch my back.
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!