What is the ruling on dealing with service offices that complete government transactions, without knowing their method of completion? And is it permissible to deal with them if they use wasta (mediation/connections) or bribery?

As long as you do not know that they are using unlawful means to complete transactions, there is no harm in dealing with them. The default is to assume the best of a Muslim, to think well of him, and to find excuses for him; as Allah Almighty says: "O you who have believed, avoid much [negative] assumption. Indeed, some assumption is sin. And do not spy." Therefore, there is no harm in dealing with them unless something appears that necessitates refraining from dealing with them.