'Get someone [adjective, perhaps modified]' is part of an object-orientated resultative construction:
- These people really get me annoyed.
- Swimming also gets me really tired.
- That kind of behavior really gets me mad.
- The [lousy] service gets me annoyed every time.
- I can't backcomb and it gets me annoyed.
- What really gets me mad is that you aren't allowed to play football.
- It gets me down when a client is a no show.
- It gets me all fired up when I think about it.
(All from the internet; corrected where appropriate.)
Some of these use it-clefts which would be awkward to rephrase, and one uses 'it' for 'this [fact]' [gets me annoyed].
But as with many strings using 'get', there are levels of informality which many would consider better avoided in formal contexts. 'Make' is often a convenient more formal replacement:
- These people really make me annoyed.
But 'mad', 'all fired up' are even less formal anyway, and 'It makes me down' isn't idiomatic.
Your example sentences are grammatical if in an informal register.