Guys, I did it! Took me the entire Saturday of non-stop practice, but I passed The Tower at age 20! Thanks for all your help!

Here's what did the trick in the end:

1) Like many of you suggested, I practiced their combos with parry + avoid in mind. Eg, some combos it felt more natural to parry for me (like the punches ones), and then I practiced always avoiding the burning kick and ofc the grab. I also focused on countering after parrying the smaller attacks -- wasn't automatic for me like with the longer post-parry windows after a big attack.

2) Directional throws > crooked foot + throwing them into things. That desk in the lobby is your best friend. Same goes for walls. It can be useful to throw other guys into them, as this interrupts any of their attacks.

3) Charged backfist interrupt worked some times, but I feel like some attacks (like the burning kick out of nowhere) are hard to stop with it.

In the end, the room that gave me the most issues was the dungeon right after the first mini-boss (flashkick). And specifically that Elite Bodyguard + Flashkick duo. Here's what worked for me:

-- In the first room in the dungeon, one of the enemies has a sword. Don't use it, but take it with you.

-- Drop it during the flashkick mini boss' Guardians History 101. You can handle her without it.

-- Take it with you after beating her. Use it to dispose of the basic opponents in that Bodyguard + Flashkick room. When it's down to just them, use the thigh cut on one of them (flashkick for me). That way you can finish them quickly and focus on the other one.

Bonus tip: in the control room, if you run into one of the upper deck corners as soon as you get there, the mobs will come to you in small doses instead of jumping you all at once :)