Some one grabs private key for address ABCD can he go after address EFGH?
No, unless your master public key also grabbed and your wallet generates non-hardened address. Most wallet generates hardened address, so you don't need to worry about this problem.
If your wallet don't use seed/HD derivation/mnemonic, you don't need to worry about it unless your wallet uses bad random method.
I am under assumption that second address EFGH has a completely different key and can not be attacked with the private key from the first address. Correct?
Yes
Note that an adversary that has compromised your system and has access to your wallet file (and managed to get ahold of your encryption passphrase using, say, a keylogger) will be able to access
all coins within the wallet, regardless of them being held in different addresses.
If only a single private key is exposed (eg. because you exported it and moved it onto a different device) than your other keys are safe within the limitations that ETFbitcoin described.