Push up, pull up, and squatting is your basic foundation. And yea, simply doing more of them will make you better at it. I've managed to maintain over 2x BW deadlift, and nearly double BW squat, simply training with bodyweight for awhile. I don't say this to impress, but to impress upon you how accentuating basics will overflow into advanced strengths.
And benching sucks. Stop it. Do dips, unless you have problems with pronated shoulders. If that doesn't hit your chest (you likely have an imbalance), try chest flies between two benches.
I work the entire body, every time. Usually 6 times a week. There's isn't a single workout where I don't perform what I call the trifecta (push up [[dips]], pull up [[row]], squat). The key is to work all variables. TUT, isometrics, negatives, blasts, stretches, planks, holds, etc. The options are limitless with bodyweight... and for me that makes it fun. Not to mention, I never, ever have injuries, nor pains or any of that shit.
And getting better at the trifecta doesn't mean you have to limit yourself to those movements. There's the close grip, chin up grip, two finger grip, wide grip, hook grip, and navy pull ups. And you can imagine the variables for push ups as well. For the lower extremity, I've been quite fond of a narrow split squat, where the hind leg is just behind the agonist leg. That and single leg deadlifts... without weight. I find using a weight actually makes it easier, granted it isn't extremely heavy.
And food, I personally have one big ass meal after my workout, and lunch if I'm hungry. I'd do a lot of research and prepping if you decide to embark on that. It isn't quite the warrior diet, or IF, but it's what I've adapted to work for me. I simply eat when I'm hungry, and eat what I know my body needs.
Lastly, don't ever count calories.