If a person is eligible to bail out, they will be given the opportunity shortly after the booking process is complete. How long that will be, after the time of arrest, varies. It depends on how long the person may be held by the arresting agency conducting the investigation before transport to jail, how busy the booking desk at the jail is, how (un)cooperative the person is during the booking process, etc.
The 10% bail is only a general rule of thumb regarding bail bondsmen. Generally, bondsmen will charge a (nonrefundable) fee of 10% of the bail amount. However, the fee is set by the bondsman and could be higher depending on the seriousness of the charge, the amount of the bail, the perceived flight risk, etc.
Behind the badge is a person. Behind the person is an ego. This is as it should be, person at the center and ego to the back.