Standard NiMh will get warm when charging at too high a rate than is good for them. The so-called rapid chargers get them very hot, whether they're new or old. The hotter they get, the quicker they'll be killed for good. I had an EverReady rapid charger once that got the batteries so hot you could barely touch them. It charged them in an hour, but they probably didn't get more than 50 cycles before they started going rapidly downhill in terms of how long they'd hold a charge. With NiMh, the slower the charge, the better. I keep my 2500mah NiMh on a trickle charger 24/7 now and charge them at 4-500 ma.
Standard NiMh aren't designed for the type of current an e-cig pulls. Don't expect yours to last very long.
There is no such thing as a protected NiMh. The chemistry is such that it is not needed. There's little or no explosion or venting hazard. They're safer than even IMR's. They might leak or even rupture, but they won't vent or explode unless you throw them in a fire, and I'm not even sure about then.
There ARE some high drain NiMh now that are suitable for e-cigs. Unfortunately, not in AAA size. Callis Kustoms has a 4.8V battery in an 18650 size that is actually a stack of four 1.2V NiMh's in series. That's the only NiMh battery I've ever seen that is decent for an e-cig. IIRC, it's only about 900 mah.