1. Will PCI-E 3.0 card work in PCI-E 2.0? Yes, the major difference is in bandwidth, but it won't affect you unless you do multiple monitors setup
2. You should upgrade your PSU before you get that GPU
3. I think you can save some money by going with a 2 GB version of GTX 760. The extra VRAM only comes in handle when display on resolutions higher than 1080p
4. You might need to upgrade that CPU cuz it might be a bottleneck
The GTX760 will work perfectly with your current motherboard.
Make sure the new power supply has at least two 6 pin PCI-E leads to power the new card.
Check the card will fi in the current case! Get out a tape measure and check.
It is unlikely the existing CPU will bottleneck a GTX760 but you should look into overclocking it, those old i5 chips overclock very well although I doubt the Gateway motherboard will allow such games.
English is my first language and I STILL make mistakes in it.