Is there any relationship between the eveness/oddness of the y (or x) coordinate and the value of the corresponding private key?
No, there isn't. You cannot infer anything about the private key from knowledge of only the public key.
A quick example. Private key 4 gives the following public key:
x = E493DBF1C10D80F3581E4904930B1404CC6C13900EE0758474FA94ABE8C4CD13
y = 51ED993EA0D455B75642E2098EA51448D967AE33BFBDFE40CFE97BDC47739922
Private key 6 gives the following public key:
x = FFF97BD5755EEEA420453A14355235D382F6472F8568A18B2F057A1460297556
y = AE12777AACFBB620F3BE96017F45C560DE80F0F6518FE4A03C870C36B075F297
As you can see, both x coordinates and both y coordinates have opposite signs.
Is there any way to know that the private key is more or less than n/2 while not knowing the private key itself?
Again, no.
Is there a way to find out the public key of an address that never sent Bitcoin but only received it?
Depends on the address. If the address is a hash of the public key, such as in P2PKH or P2WPKH, then no. If the address is not a hash of the public key, such as in P2PK or P2TR, then yes.
The other option is if the public key has been revealed via another means, such as a signing a message, openly being shared, or being leaked.