How long are you waiting for it to get cold? It takes a day or so.
If you have AC power and the auto/gas switch is on auto, it will use the AC element; if it's on gas (or you don't have AC power), it will use propane. A failure of the propane burner to light will cause the check light to turn on and obviously it won't cool. Failure in propane mode can be due to many causes, but one that's perhaps easy to overlook is a little gas shut-off valve on the burner assembly. (If your fridge was part of the recall and had the recall work done, you need to remove the sheet metal protector over the burner to get at this shut-off valve.) The valve looks like a large setscrew on the side of the assembly, and turns a quarter turn to open or close.
The light not working could be the bulb. If it's OK, my guess would be the door switch is not making good contact like it should. The switch snaps in and out of the light unit (after removing the one wire that is attached to it), and you can either order a replacement from somewhere or, if you're feeling a bit handy, see about taking it apart and cleaning/polishing the contacts inside.
The switch (assuming it's like the one on my Dometic fridge) is held together by two or three plastic posts that are welded/upset; it can be pried apart at the seams with a little bit of effort. Inside, the lever arm moves a little metal rod along an arc; this rod makes contact with hook-shaped prongs on the two contacts that extend out of the switch body. Polishing the hooks and the rod with baking soda and an old toothbrush and rinsing and reassembling was what it took to get mine working decently again. Time will tell how permanent this repair is, but so far it's been successful. The switch body stays put together well enough without the rivet things being re-welded/re-rivited once it's snapped back into the light unit.
Do check to see if the fridge is part of the recall by looking up the serial number on the Dometic website, and if the recall work hasn't been done, see about getting it done. An authorized repair place will do it free of charge.