I'm an electrician by trade and going back to some of the earlier posts, I agree that not all electricians are smart. Construction/Installation electricians aren't as generally well versed with the problems and fixes as a service electrician would be. I work more in the service end of things, and often in commercial and industrial settings.
That said, the guys that decline a job because they're not sure (ie that generator) are smart. They know their limits, and are willing to admit it. Be afraid of the cowboy who just does it anyway. never ends well.
The other guy that turned down the job even though you'd dug and laid the pipe, probably just wasn't interested because the job was now too small. I can't make money if I have to field calls, drive out, look at it, bid on it, then pickup materials and go back and install it for only a couple hundred bucks. Its just not worth it unfortuanately, can't keep the machine turning so to speak. However, in my jurisdiction at least, a homeowner is allowed to pull a permit and do their own electrical work so long as the follow the codes.
Now back to the original topic - the warm plug.
There are several reasons, and fixes, to get to the bottom of this. Most suggestions have already been covered, so this is just confirming and agreeing.
1) turn off the power and pull out the receptacle. Ensure the wires are in fact large enough (should be #12 awg copper). Make sure they're under the screws and not stabbed in to the holes in the back (I've received so much easy money from this one)
2) Check for signs of heat on the wire where it connects to the receptacle, and where any splices are made in the box. Discoloration and added or reduced "shininess" compared to the rest of the wiring is a sign of trouble. Cracking or flaking insulation is a sign of long term heat damage as well. Copper being a bit darker and not shiny is NOT an issue. natural copper oxidization is ok, so long as it isn't combined with signs of heat damage in the insulation.
3) Replace the receptacle. Go to a proper electrical supplier and buy a good quality plug from a large manufacturer. Ask for "commercial grade" or "spec grade" or "hospital grade" (local terms might be different than here). It will cost $6 instead of $.89. But will have a nice metal back strap, feel absolutely solid, and the internals will be significantly stronger.
4) Check your cord end - does it look good? Old cord ends with metal fatigued pins can cause grief as well showing signs of heating in the receptacle that aren't the receptacle's fault. If in doubt, buy a good cord end (again, not dollar store. go to an electrical supplier, and buy a proper 20A cord end). Sometimes 15A cord ends show up on manufactured devices due to the regulations for manufacturing being quite different than the regulations for electrical installers (ie your vacuum cord being warm - its anticipated that the duty cycle is so low on the vacuum cleaner that it will be ok. - commercial units don't warm up quite the same)
That should get you fixed up. If you replace any receptacles/cord ends/etc make sure you re-connect it correctly. The "Neutral" side and the "Hot" side can not be safely interchanged on the receptacle or cord ends. (swapping ends could energize the light socket so you get a good wake-up blast when changing bulbs)
Hope that helped!