The 7 pin SPI OLED will work. Make sure it’s an SSD1306 or you’ll have to use @Mr.Blinky 's homemade package if it’s an SH1106 or something else.
The Arduino UNO can work, but the Arduboy is designed for an ATmega32u4, so you may experience slowdowns and some other stuff. I think someone made a library to help with that, you can find it somewhere on the forums here. I’ll add it to my post if I can track it down.
Ideally, you want a board which has an ATmega32u4, so something along the lines of an Arduino Leonardo, a Sparkfun/Arduino Pro Micro, etc.
An Uno probably wouldn’t cause slowdowns, its CPU - the ATmega328P - uses the same instruction set architecture (AVR) and runs at the same speed (16MHz).
What would be an issue is that the ATmega328P only has 2KB (2,048 bytes) of RAM compared to the ATmega32u4’s 2.5KB (2,560 bytes), meaning that games that use a larger amount of RAM are liable to crash or potentially not even load properly.
Everything seems okay. The screen appears to be an SPI 4-pin one though. Something I’ve seen other companies do is put images of their other products (in this case, the 7-pin OLED). I think you need a passive buzzer as well, rather than an active one. It also has to be piezoelectric (IIRC).
No, you need pasive piezo buzzers (option 1407 pasive) Note that pasive piezo buzzers don’t have a polarity symbol.
as for the kit. you may want to look for the seperate parts. as the buttons in that kit ar 10-12mm tactile buttons which are quite large and the included breadboard is a half size one. You wouldn’t be able to put all the buttons on the breadboard.
with pro micros you have to solder the pinheaders and it takes up space on the breadboard. also it doesn’t have the 2nd speaker pin, RGB green pin and OLED CS broken out. If these things aren’t a problem you can go for the pro micro.