The main issue here is the geometry of your windows; while 500' below the surface isn't that far, it's still a considerable amount of force over a large area. Your city, from a quick-and-dirty engineering standpoint, is just a series of pressure vessels.
So let's say you want a comfortable 1 atm of pressure inside the city. In reality, you'd want a higher pressure inside, to help counter-act the pressure on the outside. According to various groups, it looks like divers (breathing air) should only go down to 50-60 m (~164'-196'), somewhere around 1/3 the depth of your city.
Engineering toolbox has a nifty thick-walled pressure vessel stress calculator. If any of these stresses exceed the stresses of your "glass," your connecting tubes will break! Oh, we should also mention that there are different types of glass: borosilicate glass (pyrex), common fiberglass (E-Glass), S-glass, soda-lime, even Star-Trek inspired magnesium aluminate, and many others! Each of these have their benefits and weaknesses. Let's assume you're using a 'safety glass', which is supposed to withstand 15000 psi of pressure. Using the tool from engineering toolbox, the minimum thickness allowed is: 4 inches (otherwise the hoop stress is too much). This has no safety factors involved, so you likely want this thickness to be much higher, at least 8 inches.
You can perform a similar calculation for your domes.