Can public key help attackers to break RSA encryption faster?
Can really good attacker(hacker) use public key for his advantage and break RSA encryption faster?


No; we always assume that the attacker knows the public key (that's what "public" means, after all).


  Is there some advantage to have each RSA key with different bit size?


No; the most efficient approach known to attack an RSA key is to use the Number Field Sieve algorithm; which that approach, the attacker picks one RSA key, and attacks it.  It doesn't matter if the keys he selects from has different sizes; he's likely to pick the smallest and/or the one that protects data of the most value.