Jamming and trying shit out is not the same as composing, it’s usually two different things/activities, but they can of course be combined. The counting of notes part is completely eliminated once you can just hear the note intervals, and find some basic shapes like major or minor triads or whatever. The most important thing is that you think it sounds good, not necessarily why it sounds good, imho.
Roman numerals refer to which chords are within a certain key (numbered 1-7), with special “chord symbols” noting what type of chord it is. Capital or lower-case means it’s either minor or major, etc. The usage of Roman numerals to write down chord progressions, has the advantage that you don’t have to write what key you should play them in, you can use them in any key. The distance/intervals between the notes in the chords will be the same.
All of this can be referenced through sites like this. There’s no reason to learn this shit by heart if you’re sitting and composing stuff, but I’d recommend having some notebook or thing to write shit on, on the side, to have something to reference yourself for later use.
This is exactly what I wrote here:
It’s a very iterative process, but in time, you’ll get more familiar with how to position your hands on the keys, and develop a sense of how far different intervals are from each other. When you feel comfortable enough just doing this, then you can start thinking about scales and whatnot.
Regarding this bit, I’d say it’s good for just practicing to play on the instrument, but it’s not very useful for ear training, since everything you play can sound “correct”. It’s good for motivation though, I can see that. I usually just put on some 10 hour Youtube piano jazz video and play along on my guitar, trying to anticipate the chord progressions with melody notes etc, ear training and guitar playing knowledge slammed into one activity. I usually do that for half an hour or so, a couple times a week, when I have time and mental energy for it.