Hi, I'm also new, and I haven't found anything that teaches here. But I can give you some ways you can teach yourself.

As for intervals: I suggest you learn songs for each interval or specific melodies. This way you will recognise them easily.
This video has songs for each interval.
Also singing intervals really trains your ears.

If you want to identify inversions, you first need to identify intervals. Once you can tell the difference between fifths and fourths you can try my method:
1. Listen to the chord played and sing the three notes out loud.
2. If there is a fifth between the lowest and the highest note the inversion is in it's root state.
3. If a fourth is between the middle note and highest note it is the 1st inversion.
4. If a fourth is between the lowest note and the middle note it is the 2nd inversion.

As for writing down melodies, I do that by identifying the intervals between each note and then writing it down.

Not sure if I have everything here...probably not. Feel free to ask me for more help. :D

There is actually a learning program called 'Music Theory Basics' in the learning section of this site. It sounds like that might potentially be what you're looking for. Hopefully this link works. https://www.tonegym.co/program/view?id=WE234BSY&ut=school

There are also many suggested music or music theory related videos in the learning section of this site. In general, I've found youtube to be pretty good source for music theory related topics. Might have to do a bit of searching, but there's a wealth of information out there.

You can learn from our Learning center but please keep in mind that ToneGym is aimed at ear training and not music theory education.
https://www.tonegym.co/dashboard/school

Check out the stuff under the toolbox icon (to the right of "Contests", at least in my UI) for free-form exploration of intervals and other stuff.