QWill this spell's automatic ability trigger only while it's in my cemetery?

AYes, it will.

QIf I have a Delta Cannon in my cemetery, and I summon Tetra, Sapphire Rebel, in what order should I resolve their abilities?

ASince both abilities are triggered simultaneously, you may resolve them in any order. However, if there are 4 cards in your EX area, the order in which you decide to resolve them determines which card is put into your EX area. If you decide to resolve Tetra, Sapphire Rebel's Fanfare ability first, you will put a Repair Mode into your EX area. Then, since your EX area is full, you can't put Delta Cannon into your EX area with its automatic ability. Likewise, if you decide to resolve Delta Cannon's automatic ability first, you will put it into your EX area. Then, since your EX area is full, you can't put a Repair Mode into your EX area with Tetra, Sapphire Rebel's Fanfare ability.