I agree with pretty much all of Pull's comments.
A few words about OHP and push presses, as you say you want to work on these.
At the moment, I'm recovering from various injuries that prevent me from bench pressing and deadlifting. I've become interested in increasing my overhead work and have been training this three times a week.
Over about a month, my strict overhead press has gone from 200 to 221. I'm pleased with this, though this is recovering strength to levels held a couple of years back rather than gaining strength for the first time. But I think it shows there's benefits in high frequency pressing.
However, there's a big caveat. I think overhead work is very demanding - more so than any other upper body exercise. It's hard on the support muscles. Working anywhere close to my limit, I find it really tough on the lumbar region - backache sometimes follows. I can manage this at the moment, because I'm only doing one or two other exercises. I would not want to combine this frequency with any more exercises than that.
Also: for me, little things seem to affect my comfort and strength on OHP. Wrist wraps seem to help - they allow the bar to sit slightly higher on the shoulders. Heeled weightlifting shoes help too. Most surprising for me was the effect of standing on a solid floor (as opposed to rubber mat). We have some wooden oly sections sunk into the matting. Standing on that felt er...more solid. None of this stuff substitutes for raw strength, of course
Good luck with your goals on this. As I've said before, there are very
few gym-goers who can get their own bodyweight overhead, so it's a great one to aim for