1 - A question for 13th Agers or 4e-ers about immersion
So, how do you stay feeling like you're in character or doing what your character would in combat instead of feeling like you're running through a checklist to see what you should do next? (Or would a real-life character in combat just have a mental check-list like an NFL quarterback and go on auto-pilot?) Does it work to just try to focus on visualizing, or to not consider the whole range of powers? Is it just a matter of practice?
My advice: when playing older versions of DnD, I always thought about what my PC would do in a specific situation. As a martial character I did not have many written-out options, though, besides "I attack with my sword" or "I charge and attack with my sword" or "I move around this opponent and attack with my sword". Of course, I could have tried to pull cool stunts, the result being determined by the rolling of dice as prescribed by the DM. Spellcasters would read through the spell description and think about how effective a certain spell would be. They could also use the martial attack forms, of course.
In 4E and 13th age, what has changed is the amount of written-out options for martial PCs. And that is basically it. It does not hinder you to do anything else you think would fit the situation. Instead, it provides a solid mechanical framework to help you narrate certain things you want to do. So the Paladin in 13th age can use Bastion and say "I try to save the Mage next to me from harm from that blow" etc.
You get better at narrating this the more you play. I am sure you will find that 4E and 13th age make narration very easy.
2 - A question about 13th Age/5e and DoaM
Maybe I'm just thinking too much about it. That another player's effectiveness depended on whether they had an odd-miss or even-miss also didn't help me picture it as more than plusses and minuses. That player did do a nice job of narrating what they did (after they saw the die result) though.
I bolded the important part. DOAM is not a problem unless you take "miss" and "hit" literally. "Hit" and "miss" are abstractions, just as "save" is. Because when you "make your save" in older editions of the game, a lot of times, you still take damage (and are therefore not "saved" at all). And since a combat round is 6 seconds of action, swings, moving, ducking and hp are not considered meat, there is a lot of room to narrate DOAM. Obviously, you play with people who are able to narrate things in a plausible way. That is important.
Oh, and about the "x happens on an even/uneven roll" in 13th age: just describe what you want to achieve with the power you are using. Pick a fitting narrative for the result that could be achieved by using that power. If you roll even (or uneven, depending on the power description), you succeeded, otherwise you did not. So you still deal damage, but there is no extra effect.
Have fun playing 4E/13th age.