My mini fridge is not working. Should I repair it or simply toss it out?
Well, even if they find the problem they are still going to charge you 30$+parts+install time= a colossal waste of time and money. I've owned a numerous array of mini fridges and none of them last much longer than 2 years, the workmanship on these just isn't worth fixing. 

Unless the fix is something you can readily check, like a reset button, I'd cut you losses and buy a new fridge. Also because the compressor started and then died, so it sounds like a blown compressor, which is not worth replacing.