Just how strict are airlines with the bag that my c-pap goes in? It came with a special one, but can I put it in a different bag? Can I then use the additional space in that bag for packing more stuff?
According to the TSA website, you have to take the CPAP out of the bag for it to go through the X-ray. The tubing and mask can stay in the bag.
To me, this says that you can put it in whatever bag you want. However, there's a caveat to that. Medical equipment does not count as a carryon. So if it is in its own case, it won't count as a carryon, and you can still bring whatever other carryon you are allowed to bring. If you put it in another bag and put other stuff in it, my guess is that it will likely be considered as just a normal carryon and you may not get away with using that as a way to get an extra carryon. (I use my CPAP bag to stuff extra stuff in all the time, though haven't done it on an airline - but my bag has lots of extra room when I travel because I don't bother to take the humidifier with me when I travel to save space.) So I would just leave it in the normal case, take your normal carryon, and take a strap to attach the CPAP to your other carryon to make it easier to handle things.
A couple of other hints: most places recommend taking a letter of medical necessity or your prescription with you when you fly with a CPAP. I've only flown with mine once, didn't take one, and had no problem, but others say they have had issues before. It comes down to whether or not the TSA agent knows the rules or wants to play by them or what. But the recommendation is to have the letter from your doctor just in case.
Put the CPAP in a plastic bag inside the case, so when you take it out and put it through the X-ray, it isn't exposed to the germs, handled uncovered by the TSA agents, etc. TSA actually recommends that on their site even.
Put a tag on the case that it is medical equipment. Some of the CPAP supply places have them that you can buy, but I just made my own. Do a search of "CPAP medical supply travel tag" and you'll get hits for some to buy, which will let you either buy one, or give you good examples if you want to just print up your own.