Weird MacBook Air 13" not starting

The connector in your third image is not used, as such you can fully remove it. Given its location its not uncommon for it to get messed up causing issues! Carefully desolder it and make sure the pads are not shorted. I go one step further, I paint it with a clear lacquer to prevent the area from corroding up again.

I would start there. And no the system won’t boot as it doesn’t have a drive with an OS present. It will get partway with a blinking Question Mark in a folder icon. If you have a bootable external drive connected you should be able to boot from it. If not there is a deeper issue here (after you fix the connector issue I first stated)

The SSD blade drive mount screw mount is missing! The best you can do here is get a block of plastic cut to the correct height and large enough to glue to the logic board with out getting in the way of the components on it. Again that’s what I have done and once mounted and cured I then carefully mark the spot the screw is needed and drill it and tap it or you can press into the plastic a screw with a soldering heating it so the plastic melts a round it.