If you are shooting in fully automatic (green box mode dial), the flash will only come on when the camera thinks it needs additional light for a correct exposed image. If you want to force the flash to fire you will need to use a mode other than Auto (I'm not sure if you can force flash on in P mode or not).
A great wide angle lens is Canon's EF-S 10-22. There are less expensive 3rd party lens but I don't have specific knowledge of them. Keep in mind when shooting with a wide angle lens, it's easy to distort objects when the subject is close to the lens. I understand serious architectural photographers use tilt-shift lens to manage distortion.
The T3's built-in flash only covers up to 17mm. This means if you have lens wider than 17mm, the flash will not reach the sides of your images. You might want to purchase a separate flash to provide more power and better coverage for a wide lens. A real benefit of a separate flash would be to bounce the flash off the ceiling or a wall and have the ability to increase/decrease flash output. The 430EX II flash covers a little wider, up to 15mm and has the ability to bounce the flash. The 600EX-RT has even more power and or course can bounce. I'm not familiar with the 320EX but it also has a head that can bounce flash.
It should work in P mode with either the built in flash (which you may need to make pop up) or a Speedlight mounted on the Hotshoe. The camera will pick a speed up to 1/200 sec depending on the natural light available. Agree on the 10-22 & the fact that the flash needs help to cover the extra wide angles.