My house has a Goodman split-system heat pump with an AprilAire
humidifier. The system is about 8 years old. The inside unit is
in the basement. The outside unit is just outside the basement,
on the north side of the house, about 20' from the inside unit.

Typically, the inside unit starts up about 10-15 seconds after
the outside unit. There is a humming noise coming from the
inside unit during this delay, but it goes away once the unit
starts up. However, I've noticed that occasionally the inside
unit won't start running. It will just sit there, making the
faint humming noise. I don't know how long the system will run
like this, because when I notice it happening I shut off the
system, let it sit for a while and then turn it back on. I'd
guesstimate that it has "run" like this a couple of times for at
least 15+ minutes before I noticed.

I've only noticed this behavour in the last few months, and only
seen it happen 4 or 5 times. I have no idea how often it may
have happened when I haven't been home.

There are two copper pipes coming out of the inside unit (I assume
they go to the outside unit). One is about the width of a pencil
and is uninsulated. The other is about 1/2" in diameter and is
wrapped in some kind of foam-rubber jacket. The thin pipe is
run alongside the thick pipe, with either duct tape or a plastic
tie every couple of feet holding them together. When the system
is off, the copper lines are cool/cold. When both units are
running, the lines are warm. When just the outside is running,
the thinner line becomes VERY hot. It is so hot, it has melted
the outside insulation of one of a low-voltage line that it is in
contact with. I don't know the temperature of the thicker copper
pipe when this happens.

If there is a problem, and I call in an HVAC repair-guy, is this
enough info to diagnose the problem and fix it, in spite of it
being so intermittent?

Over the summer, I had the opposite situation. The inside unit
ran fine, but the outside unit didn't run. In that case, it was
obvious there was a problem because the outside unit would not
start at all. It was fixed in about 10 minutes when the HVAC guy
replaced a capacitor (about the size of an 8-10 oz drink can).
Does the inside unit have a similar capacitor that could be on
its way to failing and causing this intermittent behaviour I'm
seeing?

thanks