I've lived in the GCC for 10 years. Are there bugs? Yes. But the only cockroaches I've ever seen in any of my accomodation were brought in by my cats for entertainment purposes (crunchy on the outside, chewy on the inside) so all I ever saw was the odd carcass.

Most compounds and apartment blocks have a monthly spraying for pests. In my accomodation, I only allow them to spray the drains because of my cats. So far, so good. Take your trash out regularly to the outside dumpster and keep your kitchen clean (you may want to keep things like flour, rice, cereals, etc. in a plastic container rather than the original packaging) you'll be fine.

Split ACs commonly have this problem (old or new) because they're not installed properly - keep on top of the technician if you do have a problem (and don't put any furniture underneath them) and you should be able to avoid or prevent it from happening.

The only other house critters I've had are the GCC equivalent of a gecko. They're really cute and they eat bugs - they are also a source of entertainment for my cats but they are rarely caught. They can startle you when they move but they are not a hazard and will help get rid of any unwanted creepy crawlies.
Signature line > "You can't fix stupid"