I was just told that heat and emeralds pose a problem for cracking.

Among other things. Alot are treated also which is a whole nother mess.

Is there a chart or rule of thumb with respect to which stones/ gems can be
heated safely, and which cannot?

Thats a tough one. I have a chart somewhere but don't seem to be able to
dig it up. Generally you can heat diamonds even to the point of retipping them
(gold not platinum) with the stone in place assuming they are not treated.
Sapphires and rubys take a good amount of heat as long as they are not
treated. I'll usually even retip with the stone in place if it's not a big or
expensive stone. Repairs in the vacinity of the stone are generally ok.
Emeralds don't take heat well and are usually expensive. Generally you
don't do repairs with a torch anywhere near the stone.
As a general rule I keep the torch away from anything other than diamond,
sapphire and ruby.

I understand that pearls are heat sensitive... would it be fair to say all
"biologically created" gems cannot be heated (ie, pearls, amber)

Fairly good rule of thumb with the possible exception of petrified wood
(never tried to heat it). They start to burn at very low heats. Being porus you
also have to be careful what you heat sink them in.

ALSO -
How do I remove a stone from a bezel mount? Is it a process of gently
peeling the bezel back?, or should I cut the bezel away?

If you are replacing the stone I'd break it out and then gently push open
the bezel. If you are saving the stone (and it's set properly) you generally
will have a better chance if you cut the bezel. You can try to push it open
with a graver but you tend to tear up the bezel and damage the stone. If it's a
cheap stone, and a standard size, you may find it cheaper to break the stone
and replace it rather than try to push back the bezel. TELL THE CUSTOMER if you
plan to do that.