Let's say i have a practicable quantum computer or other device capable of rapid factorization of large enough integers.
What are the consequences to a developed bitcoin network?
Any way it could let me cheat in generation?
Any way it would let me cheat in transactions?
If you had any system, quantum or not, that could solve the discrete logarithm problem, yes, you could generate the private key from any known public key. With that you could steal any coins you want.
However, you could also probably steal most of the Dollars, Euros, Roubles, etc.