You may feed a 14-50 outlet directly from your main panel since there are unused breaker slots available. Use the correct breaker of course, and correct conductor type/size for the load. Do not forget voltage drop and temperature ratings when choosing cable.
You do not need to add another sub-panel for a single outlet.
Do not expect the code book to tell you what you can do. It's there to tell you what you can't.
Wow, very mysterious.
First of all, if the AC unit trips a breaker, it may be the compressor is intermittantly having a hard start. Mine was doing this in cooler weather, or at night. Adding a hard-start capacitor to the compressor fixed it.
But your event looks more troublesom than that, as you have unrelated circuits tripping together. What I would first guess is that you are having brown outs. The brown outs could be caused by the AC compressor having a hard-start. Or the brown-out could be caused by the grid voltage being low.
When a AC compressor starts, it acts like a dead-short for a brief moment, until the motor is spinning. The unit starts in the "locked rotor" condition, and the locked-rotor current is usually many times that of the breaker, so a unit that has a 40 amp breaker might have an locked rotor amp (LRA) rating of over 100amps. Most breakers trip on heat, so they can take a load 3 or 4 times higher than they are rated, for a few miliseconds.
So if your AC is slow to start, it might hit your house with a 100+ amp load for a second, causing your house voltage to drop. If the voltage drops, then many things you have plugged in with reactive loads will increase in current as the voltage drops out. The breakers run off current only, and don't care that the current is caused by a brown out, so if the brown-out causes current to exceed ratings, breakers trip.
So the first thing I would do is tell your AC repairman you are popping the AC breaker from time to time, and ask him to investigate putting a hard-start kit on your compressor. What the hard-start kit does it ads a little extra boost of start-up power to the compressor, so a tired old compressor can start easier, and avoid tripping the breaker.
Replacing the AC bereaker may be called for, but don't put one in that is larger than the name plate rating on the compressor. The compressor will say on the nameplate how large a breaker you should use. Don't put a larger one in, a larger breaker will not make a hard-start compressor better, it will make it burn-up quicker. The only fix is a new compressor or a hard-start kit, never a larger breaker. Of course that assumes it is your compressor. It could be something else entirely, like a power grid brownout.
In my case, the compressor hard-start was only when it was cool outside, but that is just one odd case, most of the time a hard start compressor will hard start all the time, or only when hot, or only when the unit is short cycles, or only when it's been off for a while. Your AC guy should know to look for hard-start when you mention the breaker tripping intermittently. You should hear him talk about trying a hard-start kit even if you don't mention it to him, but that doesn't mean you can't suggest it to him anyway... if he thinks you know what is going on, he is less inclined to give you B.S... If he is unscrupulous, he may suggest a compressor replacement first, but really a hard start kit is a better first move if a hard-start condition is suspected.
Best Answer
Certainly sounds like you have a wire/outlet that is connected to two different circuits, given your symptoms.
First - when dealing with something whacky like this, just shut off the main breaker if you need to, say, cut it. If it's still hot with the main off, you have some really interesting wiring going on. And if, for some reason that should never really occur, you have a need to cut romex live, use insulated handle wirecutters and only cut one side at a time, then cut the ground wire last from the neutral side, rather than shorting out the wires in the cable with the cutters. That was not a very good or safe approach to the problem (you could have just left the thing alone until it was sorted out - cutting it does not seem to have helped in any way, and I don't see any way it WOULD help.)
Second, see what happens if you turn off BOTH the "breaker on the main" and the "breaker on the subpanel" that you identified as being associated with it. NOW is it dead?
Then, go looking for where someone connected the two circuits together. Or, call an electrician to have a look - the life you save may be your own.