Welcome to the forum!
If I can add to Jeff's very good advice, still feeling like I'm new to training some days, even after 8 years or so.
Take your time to work on technique. Don't look at rep numbers as much as feeling the strength you are gaining. Your technique is more than likely average to good at best. That leaves a lot of room for improvement without necessarily gaining reps every week (which you won't after a while).
Look at these articles and work on the accessory exercises too. Even if you aren't working pull-ups yet or pistols, the same principles apply. You won't get perfect reps the first time, but strive in time to get those. Work on the plank, hollow body hold and hollow hang. Consider these variants of the main skill and add them into your GTG practice. For example, work on a good plank before your push-ups, think of it as working on your push-up position. Do 1-2x15 sec holds before starting your GTG practice. Same before rows with hollow holds on the floor or on a bar. Keep the holds brief but try and work on the finer points, one at a time. Work on perfecting each point one at a time while also considering the whole. One week focus on squeezing the abs strongly, one week make the glutes the focus while also keeping the abs etc.
One Good Rep: How to Perform the Perfect Push-up | StrongFirst
One Good Rep: How to Perform the Perfect Pull-up | StrongFirst
One Good Rep: How to Perform the Perfect Pistol | StrongFirst
I stayed stuck at 10 push-ups for years and it's only when I started focusing on having the perfect plank while doing push-ups, doing a pause at the bottom of each rep before pushing back up and
stopping as soon as my reps slowed down, that I started making progress. These principles apply to elevating your hands also. You can do that while doing them on a chair or bench or even the wall (I have for months in the past). These technical aspects will make them harder, you will do less reps, but you are getting stronger, not chasing reps. Keep that in mind. Testing every week isn't necessary. I would do ladders, starting at 1 rep and stopping when my reps slowed down. Sometimes it took me 10 minutes to do a ladder of 1 to 5. I'd do a couple of those per day, with only one big day per week, the rest being average. I slowly went from 2 ladders of 1 to 5 (30 total reps) to 5-6 ladders. You might progress much faster than me, just know that it takes time to refine your technique.
Also don't worry about doing a fixed number of daily sets, do more some days and less on others. Something like 3-4-5 or 6-7-8 or 6-8-10 can be good general templates and allows you to have easier days, but 4-7-9 or anything else could be good also depending on your available time to train, as long as you respect the principles of staying fresh and keeping reps perfect in time you will make progress.