Very close, six's monkey, but you missed that you can also play them as soon as you receive them (although it is rare to receive Door cards on others' turns, it can happen). Otherwise, you are correct that you can only play them on your turn, but can discard them whenever you feel like it.
The rules are clear on when you can play or discard these cards. For example, the Munchkin Rules state in the Classes and Races section of When to Play Cards:
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Originally Posted by Rules
These cards may be played to the table as soon as they are acquired, or at any time during your own turn.
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And in the Character Stats section, we have:
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Originally Posted by Rules
You can discard a Race card at any time, even in combat: “I don’t wanna be an elf anymore.”
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You can discard a Class card at any time, even in combat: “I don’t wanna be a wizard anymore.”
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So to fully answer Dave's question... Generally, only the turn player can fully switch from one Race and/or Class to another. The other players can stop being something, but they can't become something else (except Human) unless they happen to receive an appropriate card.