First, congrats on the new Vette. You could use regular motor oil but there is a reason Mobil 1 Synthetic is a factory fill. Synthetic is far superior to conventional motor oil at handling heat and friction. I read recenty that many of the C4s were equipped with engine oil coolers to keep the oil temps down. It was not until GM started using synthetic as factory fill in the C5 that allowed them to forgo the use of an engine oil cooler. My advise, stick with Synthetic.
As for your transmission, with 70K on the odometer, you should have an tech inspect the fluid. If it has never been changed, you could be in trouble. If it still looks new and does not smell like varnish, you will probably be okay to have the transmission flushed using one of the professional machines life BG. A simply drain and re-fill is not going to do it. Flusing is the way to go. NOTE - If your trans fluid is discolored or smells like varnish, leave it alone. Changing it could spell disaster for your transmission.
Just my opinion, but I would not recommend the 160 TStat. You car will burn more fuel and I doubt the horsepower gains are significant enough to feel. A TSTAT does not control max temperature as some might think. If you were to install a 160 TStat with changes to the ECM program allowing your cooling fans to come on earlier, that could yield some HP gains in conjunction with other mods. If you are talking about just doing the TStat alone, I would recommend against it. Good luck with the new car.