Hi Wishful Thinking,
I trained myself for many years so I feel I can give you some advice. For your handstand, do them against a wall and hold for as long as you can with straight arms. Make sure you are pressing your back against the wall to remove any arch.
For back bends, try walking like a crab in the back bend, backwards and forwards. It strengthens all the muscles you will need for back walkovers and front walkovers.
For cartwheels, always start with your arms by your ears. Lift the back leg as your body goes forward to put your hands on the floor. Arms are by your ears again when you finish. Make a line on the floor with tape to get
really confident for cartwheels on beam.
You'll need shoulder strength for that "push" in a Roundoff. You can develop this strength by doing shoulder shrugs in a handstand against a wall. Never bend your arms in a shoulder shrug. Sag in the shoulders and then extend. Do as many reps until you are tired.
Conditioning is very important. Do push ups, V-ups, releves, pull ups, crunches, handstand holds, wall sits, rocking in a hollow position on your back, stomach and both sides.
Always work on flexibility! Especially the wrists and ankles. Splits should be done with squared hips. Straddle splits with feet upward, not rolled forward. Shoulder flexibilty too! Get a dowel stick and stretch your upper body with it.
These are few suggestions to work on. In a few months, if you do these faithfully, you will notice a lot of improvement. Go for it! I'm behind you all the way!
