I am installing 3 1/4"x 10' strip flooring in a bed room that is 15' x 15' and is in an older house with a 3 1/4" T&G sub floor. The joists are 16" on center.

I know what to do when laying out a floor using random length boards, but I am a bit unsure of how to precede when all the boards are the same length. Do I start by just cutting the first few boards at random lengths and then using the off cuts to start my new rows. And then move across the floor repeating the pattern?

There will be no structural deficiencies using the cut-offs to start the next run if you have a sub-floor. If you want to have a more "random" look you can use two or three more different lengths cut into shorter starter boards that will have different repeats going across the room.
Short boards are not commonly used in the middle of the room, just for starter or ending boards, to take advantage of the look of long lengths that you are starting with.

One follow up question. What gauge cleat would be best for installing this SYP flooring 16 or 18?

Thank you

I use 16 ga. for 3/4 inch flooring.