A. I wouldn't use 'vexed' if I were you. It's very old-fashioned. As GuideDogSaint says, your sentences are grammatically ok, but nobody would ever say them. Forget about 'vexed' - it isn't a useful word.
B. Do they have absolutely the same meaning?
1) get on my nerves
2) get irritated
What strikes me here is the fact that the grammar of these phrases is completely different, with different subjects and a totally different sentence construction:
Noisy car radios get on my nerves.
I get irritated when people play noisy car radios.
As for the meaning, yes, they do mean the same.
One difference in meaning is that while you can be irritated immediately, it takes a little time for something to get on your nerves. If someone does something annoying once, maybe it doesn't bother you, but if they keep on doing it then it can begin to 'get on your nerves'.
Another difference is the situation in which you'd use them. As I'm sure you know, phrasal verbs are always more informal than their synonyms. So, here's a question for you : if you had to write a letter of complaint, which would you use?