I rate fly spray. If you know the general direction that your Buzzy friend is in, give it a good squirt of that and ignore me get rid of them. I would love to be more friendly to bugs, but as a blind person I don’t know if it’s a harmless fly or some kind of hornet that’s going to cause me issues. So I take no chances.

I must admit, once bug gets in my room, I usually must just accept it is a new resident. Sometimes I can catch larger non-flying ones, but anything else is too hard. One thing that sometimes works; if you have the type of clip on desk lamp where the shade is funnel shaped, certain bugs will be attracted to the lightbulb and fly into the shade. I used to turn off all the lights but the desk light, so that the bug would fly into the lamp, then cover the opening with a book, unplug the lamp, and dump the bug outside.

I like to suck them up with the hose attached on my vacuum.

Having screen seems to be best bet. But again, some bugs might slip in. You might wanna have a payed sighted person do the cleaning once in a month may be. Very curious to here solutions from others on this. I am gonna have this issue in few months.

I’d go for a magnetic screen that you don’t have to fit into your window. More of like a flap so you can still easily climb in and out. they work great for the flyers. Add some fly tape to the top of the window and any that do come in will likely get stuck on it. You could do some bug tape around the window ledge for the crawling ones, but if you’re anything like me, you’re gonna stick yourself to it and probably stop eventually.

Put a screen on the windows (you don't have to make one, you can buy an inexpensive one), and also put one of those plugged repellents, they are safe even for babies. Don't use spray repellent.

I have a cat with a severe hatred of bugs entering his turf.

Ones on the carpet I sued to be able to see out of the corners of my eyes (oddly better than looking straight on), butt hen they annoyingly put in darker carpet so now I can't even tell at all if any are crawling by!

For flying ones, the big ones I can usually notice when they land on a wall and get them, but something like a fly or mosquito I don't even bother trying with and just hope they hurry and die. The rare times I hear a mosquito buzzing I sometimes just like clap my hands together in random places hoping I accidentally get it. haha.

I have gel squares by doors and in coorners. These trap anything from Roaches, rats, spiders, and lizards… There are also some hanging gel strips that capture flying insects.

Mosquitoes don't fly very well so a fan can knock them off course. Maybe one pointed towards the open window or in your room/bedroom to keep them off you?

There are bug zappers that attract them and kill them but I don't think you want them close to you or in your room.

A screen is the only way to keep flies from coming in an open window. Sometimes it keeps people out that you didn't invite in from the fire escape.
If you close your window now and leave it closed for a few days, you will probably find that you still have flys from when eggs were laid, usually someplace around the window sill. One fly made dozens of flys.
The screen keeps out small animals as well, birds, squirrels and whatever mess they make. Birds bring lice.
If you are in a rental unit, the housing code probably requires a screen for the above.
I would not be using a spray indoors unless I positivity knew there was no food or eating utensils around, or open flame.
Screens also prevent children from falling out windows.
Since you said you use the fire escape, and to anyone else, storage of anything is not allowed on fire escapes. It is an obstruction to the means of egress and will hinder the escape from the building and slow access for the fire department (trip hazard).
Edited for spelling.

My cat helps me out!

A screen is definitely the best way to go for a way to keep them from entering. Once they have entered though, maybe you could try a handheld vacuum that would suck them up. I"ve also been considering getting a Venus Flytrap, but I'm not sure how affective they would be at immediately removing bugs. Bug zappers are also great.

The electric rackets are easier for me than a fly swatted because you can wave while they're in midair.

Another option is raid spray. It'll kill em and its easier to spray in a wide arc that's overkill and hit the bug.

A good question. Generally keep your space really clean and you get less regularly clean as thoroughly as you can. Some of these other suggestions are good though.