Is paying money for government employment considered a bribe, knowing that there is no intention of paying it and that it is a common method for getting things done?

If you are more deserving of the job, then paying money is not a sin. However, if you are not deserving of the job, or are equal to others, then what you paid is a forbidden bribe. The Messenger of Allah cursed the briber and the bribed in matters of judgment, and you should have refrained from paying the money. What is incumbent upon you now is repentance and seeking forgiveness if what you did was a bribe.