First, realize that with a single zone system and multiple floors, it's difficult to get the temperature correct everywhere. Also, since you mentioned this was an attic space, you're likely against the roof and getting heat from every wall plus from the computers and people in the space. In short, you're fighting a losing battle.
I've got a similar challenge, though not to your extreme, and here's what I have done. First, learn how to adjust the baffles on the ducts. In the summer, you want as much as possible going to the high floors, and the reverse in the winter. Get someone to feel the air coming out while you're pushing the baffle all the way to one side or another. For me, it made little difference until I pushed it all the way.
Next, exclude any rooms you don't need to heat/cool, e.g. a guest room or storage space in the basement. Shut the door and close the vents to reduce the effort you're putting on the HVAC.
Finally, I swapped out my return grille with models that accept a filter (they are hinged and open after releasing two small clips). These return grilles are special orders from the big box stores, but they carry them. I still leave a thin blue "rock catcher" filter directly on my HVAC blower itself. And I put a filter in every return except the upstairs in the summer and downstairs in the winter. The idea being to suck out as much air as possible from location that needs it.
This means that switching between heat and cool is a process, flipping the baffles and swapping in and out filters. But a little effort is better than a lot of sweating.
One last suggestion, since it sounds like you're running a small server farm in your attic. Either move that down to the basement, or consider replacing the window unit with a ductless mini-split system. You get to see out the window, and these systems have configurable thermostats.
Drywall is the enemy.
That dryer vent is disconcerting; lets find out what other shenanigans are going on here. You may not have ever done ductwork but at least you seem to know a proper installation will require opening the walls to install rigid (use flexible only if you must).
Can you run it in the attic, sure: at a loss of efficiency; longer run, more leaks\back-pressure. Also, efficiency will drop like a rock if you run any ducts through unconditioned space. Ideally everything is within the building envelope.
Running ducts isn't that hard; fixing what you had to destroy to get them in, is.
You will cut yourself while working with sheet metal. Proceed slowly and methodically; everything is sharp...
Best Answer
Welcome to the wide, wide world of second story air conditioning. If this was done to prevent it from being manipulated they wouldn't have installed it. This was done just to prevent the damper from closing due to air flow. Find out where this duct feeds and if it's to the first floor try closing it a bit to divert more air to the second floor. If this does feed the second floor, you might want to add a duct fan to increase the air flow. They can be controlled manually by a switch or hooked up to your air handler. Good luck.