Bigger and Stronger are not necessarily synonymous, you may want to get bigger looking, but you might not get stronger, and vice versa.
I think the term you're attempting to describe is like progressive overload.
I think what you're doing now is somewhat futile in that, you need to build your muscles, and condition them to handle more weight, more reps, etc.
The weight increase should be steady and progressive.
You may want to try and use those 25 lb dumbbells for some higher rep pyramid sets, doing things like 5 sets 20-15-12-10 and for example supersetting standing bicep curls with tricep extensions with little to no rest will create an environment for muscular hypertrophy. and muscular endurance. Both of which will help you when you want to increase the weight.
If your interest is in Body Building or getting bigger, you will want to keep the moderate-high reps (9-12) with a somewhat heavy weight. If your interest is increasing your strength//oppose to getting bigger, you will want to increase the weight to roughly 80%+ max load, and do low reps (6 or less), pushing your central nervous system into extreme acts of strength, oppose to high consistency patterned demands on your muscle.
I would say both activate a high amounts of fast twitch muscle fibers, but the high weight/low rep will activate more...
I would also not leave until you did 4 sets of at least 2 compound exercises, and then add some isolation movements to a particular muscle group for that given day of the week, whatever you want to focus on. This will create your split. If pushups are your only chest exercise, you should be doing them to failure every time and constantly increase your reps per set, even if you have to collapse in between cranking them out, same for pullups...
I was planning on sticking to the same weight each day until it became "easy", and then increasing it and repeating that process. Is this approach somehow flawed?
It's not flawed, it's just not that convenient for a progressive system, usually you want to increase the weight or the reps in order to push yourself, sometimes it's easier to do 10 reps of 200lbs oppose to 20 reps of 100lbs. It's also about time under tension which you should google. So if you can't do a bunch of push ups, you can do them slower, creating more resistance.
Are there any significant changes I should make to my program? What kind of results should I expect to see?
As it stands you are missing a vital part of your strength training, and that's your legs. Working out your legs will boost metabolism and burn more calories at the gym due to the size of the quads and hamstrings, glutes and more importantly the core. I would research a lot about proper form on two main compound lifts, the deadlift, and the squat. Without those, you are just training your upper body, and you would really be cutting yourself short.
Also, implement progressive overload system into each routine.
You could expect to see fat loss within a week honestly, depending how strict your diet is, and how much you push yourself.
Your body fat between 12-16% is probably ideal, so you're going to have to change it up eventually, probably have an active rest day with some HIT.
Hope this helps some.