Andor's Trail

The guard at the front gate, or the guard at the side gate?

the guard at the front will only take your money and laugh at you. The guard at the side (in the garden) can be bribed for letting you through once per bribe. As soon as you hand over your money just leave the conversation and go through. iirc the minimum bribes necessary are 300 gold (1x100, 1x200).

being thrown out, finding Norgothla, and being send back for evidence?

I don't have time to replay this atm, but iirc you have to bribe yourself back in (200 gold this time), get a rose from the rose garden (maybe from the guard as well, I don't remember? try around the garden), present that rose to Hannah in the tower and get her to give you the flute. The evidence you're looking for is Lovis himself, currently imprisoned in the castle. Once you have the rose, continue exploring the tower, someone there will help you get access to the prison. The rest of the quest should be self-explanatory.