I'm coming into this conversation a bit late, and I'm sure most questions have been answered, and I may have missed some things, but here are my suggestions
1- Definitely Rent a Car. I would think the cost of the BGT Tram thing and then the possible taxi rides would come pretty close to the total of the car.
2- It may be best to stay closer to the St. Pete Airport. The area near BGT isn't the greatest (but its not terrible either), but I think the biggest concern would probably be traffic. I usually don't have to take it, but I-275 is a pain in the butt!
3- I may not have lived in Tampa very long, so I don't get most of the bridge jokes, lol. traffic can suck on the bridges sometimes, but other than that, nothing to worry about.
Damo, have you and your son done Sheikra yet(or BGT in general?), or will this be the first time?