NFC - FPGA - TFT Question
Using an FPGA for this task seems to be overkill. Yes, you can connect a display to an FPGA, but you can do that with lots of microprocessors as well. Development in software is also much faster.

So unless you want to design in the FPGA for the learning experience or you already have an FPGA with some free space in your design I would go with some cheap Cortex-M3 CPU board.

Regarding NFC, yes, sure. You can use NFC to transmit images. But keep in mind that you will have a data-rate of about 2 kilobytes/second under ideal conditions. So transmitting images takes quite a while. You can cheat a little and just use NFC just to detect the transmit event and do bluetooth or wifi paring for the real data. That's what Samsung did for their S-Beam feature on several Android phones.

Regarding NFC chips: Most of them are rather complex beasts when it comes to program them. Also the data-sheets are rarely publicly available. 
The PN532 from NXP is a bit aged but you can actually buy the chip and get data-sheets for it. There is also plenty of example source-code out there, so you don't have to start from scratch. If you don't have prior experiences with NFC, that would be my recommendation.