A couple problems a week into ownership
 

Alright I own a 2004 Mazda 3 hatch. First Mazda I've owned, got it from a guy on craigslist. The guy before him did some work to it. The night I got the car the check engine light came on. The last week and a half I've been looking for symptoms and the biggest thing I've found is a weird idle pattern. I idles at around 1 k when I stop for maybe 20 seconds then sits at about 750-800 rpm. On occasion though it will hit a rough patch and drop close to 500 and kind of sputter a bit, then go back to normal. Any ideas on what I should check for that and if it would tie into the Check Engine light?

The next thing is the driver window. It does not like to go up unless the passenger switch is hit at the same time. Once it starts I can let the passenger switch go and it'll go up fine, but if I don't use the passenger window switch at the same time when I raise I'll have to try between 3 and 10 time before it decides to go up.

I'm trying to avoid a $140 diagnostic check but if worse comes to worse I'll fork over the cash and get it done

It's probably a bad fuel tank purge valve- lives under the hood. Expensive from the dealer, not too bad aftermarket. Easy to change if you can figure out the silly locks on the hoses. The engine codes may not tell you enough to be sure it's the purge valve, in fact my dealer was pretty sure it was something else when I suggested the purge valve, but that's what it was. As for the window electrical problem, no clue. Good luck with the 2004. Mine is the most unreliable and expensive car I've ever owned in 40 years of owning cars. I love to drive it when everything is right, but I call it the "do you feel lucky" Mazda because getting to the destination is purely a matter of luck.

Definitely have the CEL checked before just trying to figure out what the issue could be. Most auto parts stores will read CELs for free with their scanner if you ask. The gas cap is generally a P0440, P0441, P0446 for incorrect purge flow and gross leak. Most places will ask if you've recently gotten gas, or checked the cap before bothering to scan.

Also try messing around with the wiring harness for the MAF sensor while the car is idling and see if that alters the idle at all. The 3 commonly gets damaged or broken MAF ground wires; especially so if an aftermarket intake was installed and the harness wasn't unhooked from the battery box. When the engine torques it pulls on the wires and damages them.

First things first though, check the CEL.