"impassive" sounds like a good fit. Such a person isn't exactly immune to embarrassment but they are immune to letting it show. "poker-faced" refers to being deliberately impassive and might fit too. An "imperturbable" character (already mentioned in an answer by dangph) is unlikely to show any embarrassment in the situations you've mentioned.
- "impassive" - not revealing or affected by emotion, reserved
"impassive" - not revealing or affected by emotion, reserved
- "poker-faced" - refers to being deliberately impassive.
"poker-faced" - refers to being deliberately impassive.
- "imperturbable" - not easily perturbed; calm; unruffled
"imperturbable" - not easily perturbed; calm; unruffled
