Can really good attacker(hacker) use public key for his advantage and
break RSA encryption faster?
No, the standard attacker models in Cryptography are already way stronger than this assumption and they assert us that you can't break (properly done) RSA encryption given those advanced capabilities (which includes knowledge of the public key).
or it is not important if attacker has public key?
Obviously it's harder for the attacker to break a cryptosystem of which he didn't ever see anything, but in general the name public key is chosen for a reason. By assumption it is always public and thus there's no non-negligible harm in giving it to an attacker.
Is there some advatages to have each RSA key with different bit size?
Except for the varied levels of theoretical security and the varied amount of data that can fit into a single block that come with different key sizes, there's no advantage to this. However, these days, hybrid encryption is mostly used which is why sizes at around 2048 bit are totally fine for that.
If attacker will break one of these key all other keys are also
imediatelly know for this attacker or he must start whole process of
Yes and no.
There are crypto-systems where breaking one key using the proper method essentially gives you all the other keys with the same parameters with negligble cost (I'm talking about NFS attackable DLOG groups here, using Logjam). There are other cryptosystems where you will experience a "nice" speed-up from breaking multiple keys (I'm talking about generic DLOG groups here), in that breaking the $i$-th key gives you a speed-up of about $\sqrt i$.
For RSA specifically, there is a slight speed-up that can be gained by targeting multiple keys, but it is not nearly enough to make breaking other keys "easy" after having broken one.