US and Canadian airlines are required by their regulators to accept the "medical only" bag as extra to the carryon allowance (it is actually technically "mobility aids and other assistive devices"). It derives out of their duty to accommodate disabilities.
From US DOT's discussion re: DOT rule (Part 382) (
http://airconsumer.ost.dot.gov/rules/Part 382-2008.doc available via DOT's website
http://airconsumer.ost.dot.gov/ACAAcomplaint.htm)
382.121 What mobility aids and other assistive devices may passengers with a disability bring into the aircraft cabin? Passengers may bring manual, but not electric wheelchairs, other mobility aids (e.g., canes, including those used by blind passengers), and other assistive devices (e.g., POCs), as well as prescription medications and any medical devices needed to administer them (e.g., syringes, auto-injectors), as long as they comply with applicable safety, security and hazardous materials rules. These devices and aids cannot be counted against the airline’s carry-on limits.
The key is that if those mobility aids and other assistive devices (e.g. medication, nebulizer, CPAP) need to be in a bag, the bag can ONLY contain "mobility aids and other assistive devices". Don't go putting your swimsuit in that bag. Also, of course, it needs to be carry-on size or it won't fit

I suppose one could have multiple medical bags if that was what was needed.
And yes, the airline can ask to inspect the contents of the bag to verify there aren't other things in it. i have never had an airline try to exercise that right, but I have had them verify verbally that it only contains medical things.
I have enough medical things that the extra bag is necessary for me. Between the nebulizer, distilled water and vinegar for disinfecting the nebulizer (I limit the amount to what I may need in travel as long as I know I will be able to buy more of same at destination; when that is not possible (have you ever tried to buy distilled water ? it actually can be quite a challenge sometimes) I carry what I will need for the whole trip), and a whole bunch of medications, they take up a small softside rolling carryon on their own. If I did not have so much, I would try really hard to incorporate the things in my normal carry-on, as carrying three bags is at times a chore.
I travel with an electronic copy (scanned PDF) of my most recent scripts, just in case (mostly in case I have to get replacements, as it will make it easier to transfer or get a new script). However, the pharmacy labeling on the meds has always been sufficient. As someone else noted, the TSA is not the DEA (or Customs, or local police). Their concern is SECURITY. Sometimes my pharmacy puts the label on the inhaler box; I have asked them to please stop doing that and put it on the inhaler itself; that way wherever I am it is clear the things are my Rx. If needed, the pharmacy can print you an extra label to put on the inhaler if you are using an inhaler that doesn't have it on it -- just go to the pharmacy and ask (bringing the inhaler is probably a good idea).
Finally, for the nebuilizer I spent the money (insurance did not cover since I already have a standard compressor neb) and bought a portable neb as it is so much smaller and lighter and easier to bring with me. e.g. I bring it with me into the parks. The one I use is the Omron MicroAir. Pari makes another good one too, the Pari TREK S, which has a battery option. Which one is most appropriate depends on a combo of which medications one is using (there are some that cannot be used in the Omron since it is ultrasonic) and personal preference.
Also, traveling with an inhaler that is several months old is not a problem. Rescue inhaler, for example, ideally don't get used that often -- mine will almost always expire before they are fully used. So I may travel with one whose Rx is over a year old !
-SW