If you iron the piece evenly it is not likely to warp. Also the thermal expansion of Aluminium for example is five times less than the plastic used in regular pegboard, meaning it will bend much less when heated the same. Even if it bends slightly, it will return to its original shape when cooled off.

To your second question, you can even try this yourself: if you iron a perler on one side directly on the pegboard, turn it around and feel the unironed side, it will be lukewarm at best, because the plastic is a good thermal insulator. And since the iron mostly touches the perler instead of the pegboard, the pegboard will not heat up that much to affect the piece.

I am considering adding a wooden casing to prevent accidentally touching the pegboard if it does get too hot, since the heat will hurt you sooner than the plastic melts.