What I am wondering: as you learn more do you become more aware of what the airplane is doing (and what you are not doing right)? Or did I just have an off day?
The answer to your question is yes. As you gain experience, you are better able (and quicker) to determine what's not looking/feeling right and what is the proper correction.
For the V
y climb, once you have the correct pitch angle figured out to obtain the correct indicated airspeed, trim for that and then use fingertip pressure corrections. Re-trim as needed.
Once you're properly pitched and trimmed for V
y, take a look out front and slightly to your left side to see what the sight picture looks like for that pitch. Memorize it. This becomes your visual cue for gaining V
y +/- a few knots/mph.
Your pattern altitude might be attributed to your airspeed in the pattern. Going to fast can aggravate altitude chants and shorten the time you have to make the change before your next action or turn.
Once you achieve pattern altitude, pitch to level and then adjust power until your at the pattern speed your CFI is teaching you. Re-trim the aircraft. Once you have done this, you will likely find it easier to maintain the altitude as you turn downwind and fly that leg.
Trim is one of your biggest friends while flying. If you find that you need to keep a pull or push pressure on the yoke to maintain speed or altitude, you're not properly trimmed.
A properly trimmed aircraft should permit removal of hands from the yoke and it maintains the desired attitude and speed.
If it helps, ask your CFI to work with you on this in the practice area. After 20-30 minutes of setting trim for various speeds and being level, ascending, or descending, you should have a better handle on the concept. And as a result, improvement on the items you queried us about.