And is there some way I can check out the gas system?
Is something plugged up?
Without removing the gas system (don't do that) its hard to tell. You can check with compressed air by plugging one end of the barrel and seeing what the flow out the gas tube is like, and if there are any leaks.
Honestly, i doubt that is the issue. you could take the tube out and check it out but there's really not much that can go wrong with a gas system other than the hole being too small or the block being misaligned. and these are very rare events on a factory rifle.
The extractor is the known failure point and is your best bet. Second to that i would say is the ammo, but if it goes bang and a bullet comes out the right end... it should extract.
Just to be thorough, make sure your chamber is clean.
Without any ammunition and with the upper separate from the lower, insert the BCG and push it until it locks, pull it out, do that a few times and just make sure everything runs smooth.
Put the lower back on and just cycle it with the charging handle. It shouldn't be too hard to operate and there should be no noticeable snags or tough areas.
Also, as an after though, you said when you poke the ejector it goes in and out. how much force does it take you to compress your ejector? All my ejectors have required quite a bit of force to depress. Not impossible to do, but not easy by any means. If you can easily just push the ejector in like its nothing, it might not be putting enough force on the case to eject it properly.
Borrowed from google, an image on how to test the ejector spring