What I currently have setup is a RPi3, overclocked with
these settings and N64 (Banjo Tooie)
seems to run quite ok (only watched the intro, didn't play anything as I had no controllers attached atm). Without overclock it's utter crap. To overclock make sure you have heatsinks installed and a case with fan (or something to properly cool the RPi). Also get an official power supply and maybe a
PiHut RPi USB hub. I have used quite some power supplies (ranging from iPad charger to AliExpress 4A power supply) and I always got the lower voltage symbol (with having 1 arcade usb controller, a USB memory stick and logitech adapter attached). Especially when overclocked. After I switched to the official power supply and the special RPi powered USB hub yesterday I didn't even see the lightning symbol at startup or while testing N64.
For power on/off I use a Mausberry circuit, which powers down the RPi safely (so no risk of SD corruption), cuts down the power to the RPi and powers it up again when you switch your button. Works really well (but don't expect any support or fast shipping from the seller). An alternative is a Powerblock (from one of the RetroPie mods/devs(?)). Does the same job. Maybe even better is his Controlblock which combines the power switch circuit with an USB controller for arcade controls (for two players). I would have bought that one if I didn't already had four Zero Delay USB controllers which do the job perfectly for me. Ultimarc stick should work as well...
Newest console playable is probably N64 and PSX but YMMV. Haven't tried Dreamcast yet.
I don't know HyperSpin, but I would recommend to start off with RetroPie (just download the image on their website and write it to an SD card with something like Etcher.io). It will work quite fine out of the box. Then customize it at will. You can also use AttractMode, so use skins that look like HyperSpin.
Of course you can also just download a fully setup image (what I guess HyperSpin is), but what's the fun in that!?
No freezes or crashes until now. I don't think there will be much difference between a PC or RPi.
Average cost? Always more than you budgeted.
I guess something like this (but you can find the exact prices yourself fairly easy):
- RPi: €30-35
- Power on/off circuit: €20
- Power supply: €10
- Powered USB hub: €10
- Arcade controls: €40-100+
- USB memory stick: €20
- SD card (fast): €10
- Case with fan/heatsinks: €6
So give or take: €150 without the cabinet.