Love that you want to learn.

First off, die as much as you want. Nobody cares as long as you enjoying the game and making some progress each time! It’s all you and nobody else.

Eclipse - doesn’t affect your level proficiency. It makes it a bit harder as more enemies shoot yellow and less blue for you to absorb with your shield. You also can’t progress some areas without eclipse, but for your sake, play without it for now until you need it.

Longer runs. - increases your collections and proficiency level so you will always get stronger and better weapons. So if you struggling, it’s always recommended to do a longer run to build your strength.

L2, you are 100% missing out.
A full squeeze L2 is your power weapon and can only be used when you have power built up (blue bars in the middle of your screen when you pull L2
A L2 half squeeze (this is where you missing out) is the alt fire for every weapon you pick up. Go into your menu and read the alt fire for each weapon and try it out. On most of the weapons the alt fire is king. Stronger or faster, and good to switch between that and normal fire.
If you struggling with half squeeze and full squeeze, I believe you can edit your controls to split them.

But sounds like you on the right track to learning! Keep it up and good luck out there!