The name Thermopylae has its origins in Ancient Greek θερμός (thermós, meaning "warm" or "hot") and the plural of πύλη (púlē, meaning "gate" or "opening). θερμός does come from a PIE root, g^(w)her- "warm", but it's not clear where πύλη comes from - it may be a loan from a non-Indo-European language. PIE does have the root d^(h)wer- meaning "door" or "gate," but it's not related (though it does show up in the Ancient Greek word for "door", θύρα or thúra).

As for Xerxes, the name comes from Old Persian Xšayāršā, a combination of xšaya (“ruling”) and aršan (“man” or “hero”). The first derives from a PIE root tkeh- "ruling" and the second seems to come from wers- "male".

So there doesn’t seem to be any relationship in either etymology or meaning between the two names.

I'm not a linguist so I'm not exactly sure what you mean. The place is called Thermopylai in Greek, from thermos "hot" + pylê "gate", due to the fact that there were hot springs near a narrow pass. The king is called Xšaya-ṛšā in Old Persian. Unless there is some way to arrive at both of these terms through gradual modification of the proto-Indo-European roots you mentioned, I don't see how "Thermopylai" and "Xerxes" could be connected.