The Berlin Defence, as employed by Kramnik to defeat Kasparov, who banged his head against a brick wall.

What about 13.e6? after Black defends the mate threat on g7, 14.exf7 draws out the king

That's good tactical thinking x-machine, but there is a counter which makes the position even...Bf6 and White's attack is stopped dead in its tracks. I would have expected your idea to have been looked at, but also analyzed deeper to find the counter. 13.Nc3 develops a piece which is more important. An alternative would have been 13.Nd2.

that was a lot of time spent in the DB there lol

I question 27.Qa3. Why? Placing you're own Queen on your own pawn stop is generally bad form, unless you can identify a specific advantage for doing so. 25.Qa3 was OK because of the action against the a7 pawn. The threat to the rook after 27.Qa3 was only pseudo in nature. I just don't see much advantage having the Queen at a3. Sorry that I cannot be more specific. It's not really a bad move. I just would not have made that choice.

It feels like you are going for the draw rather than a win. If the Queen wasn't blocking your a pawn, you could have tried pushing it to add pressure against Black's defense. It might not have worked, but it would have been a viable attempt at winning the game.

Makes sense. Black is trying to win the game at this point.

Or perhaps too late...as I pointed out earlier. Your rook would have helped protect this pawn. Regardless, the timing was off as you have identified.

Although this would be a difficult endgame, I would think that Black should win at this point. Since you obtained a draw, you must have done an excellent job in the endgame!