How does the chess brain evolve over time by rating (100-3500)

Most if it is unconscious pattern recognition, so it's hard for people to give accurate answers. Mostly they'll talk about their subjective experience.

I'd say the biggest difference makers are something like...

100-800 online rating is getting used to visualizing how pieces move on the board. Below 800 it's very common to lose a piece because you simply didn't see your opponent could move there, or your opponent offers a free piece and you don't take it because you don't see that you can take it. Of course players up to 800 learn many things, but I'm talking about what the most influential elements are from a win/loss/draw standpoint.

800 to 1600 online rating is developing good calculation habits. You're becoming aware of threats before they happen. For example if your king and queen are lined up on a file, you'll notice it before your opponent skewers them. If the first phase was not giving away pieces to zero move tactics (something is under attack but you don't notice) this phase is not giving away pieces to simple 1 or 2 move tactics.

1600 to ? online rating:  After players are good at not losing things for free they start noticing they're sometimes losing games because their pieces are somehow not as well placed as their opponent's, and so it's time to focus on positional and strategic ideas. Some strategic ideas are linked to the endgame (e.g. the isolated queen's pawn) and so endgame knowledge starts to have a functional role as well.

After that I assume it's just refining things. I've heard somewhere around master level it's useful to start being more concrete and paying more attention to where rules have exceptions. For example maybe you allow a structure that's often bad, or an attack that's often dangerous, because in the specific position there isn't a concrete sequence of moves to exploit it.

I once watched Carlsen play all sorts of VERY BAD moves in a blitz endgame... but magically... for some reason... suddenly his opponent had no good moves, and Carlsen had a simple winning idea on the queenside. Of course Carlsen understood all this from the beginning, and it's what made his "bad" looking moves actually good.

So the second phase is the Tactical Era?

Yeah.

And just to be clear... there are plenty of 800 rated players who know various things about strategy, endgames, openings, etc. And it also depends on the website and time control. 800 on one website in one time control may be completely different on another.

But I think that's the basic progression...
1. Easily seeing all the pieces on the board and where pieces can move
2. Easily seeing all the basic tactical ideas on the board.
3. Adding some positional / strategic / endgame ideas
4. Now that the player has collected all the basics, and is around master level, they learn to break many rules for concrete reasons.

What about me?

The brain gets better at playing chess

But what new features get added in every level?

Pattern recognition, visualisation, anticipation of threats, executing good plans and so on, in that order.

ScatteredWealth has seen far more than I do (I am still a patzer), so he is able to tell you more in detail about this,

And in mentality?

You just have the ability to see more in the position. For example, a 600 rated player may not see that his queen is hanging (an extreme example), but a 1200 will. The 1200, however, may not be able to understand certain concepts, such as king safety, positional understanding, and piece activity. The brain gradually grasps hold of these concepts and becomes better at performing these tasks with repetition and practice.

Chess evolution generally follows the same path for everyone (just at differing speeds):

1. Learn the rules
2. Learn basic opening principles
3. Learn how to convert basic winning endgames
4. Learn basic tactical patterns
5. Master basic tactical patterns
6. Learn basic theoretical endgames
7. Learn advanced tactical patterns/Learn positional concepts
8. Learn practical endgame strategies
9. Master advanced tactical patterns/master positional concepts
10. Learn deep opening theory

In general, by step 5, a player is typically about as strong as your typical 2000-rated OTB player.  That is where most people stop progressing as the next few steps require more effort and are perceived to be "less fun".  Also, many players reach ~1500 and try to jump straight to step 10 (which is mainly where you see strong IMs and GMs), which causes them to stagnate.

To draw a comparison:  If you watch Arnold's old workouts when he was winning Mr Olympia every year and try to mimic him, you will be frustrated that you are not seeing his same results.  However, if you looked at what he was doing when he was where you are now and did that instead, you are likely to see much better results (ignoring the PED usage).  Far too many people look at what is going on at the top of the mountain and forget that the people dancing around on the summit did not get there by dancing.

True