1) Have you tried just asking them? Even if you are essentially auto-accepted, asking implies that you care about what the other party says, leading to more amicable relations later down the line.
2) See 1. If you'd like to use a base, ask
3) As I understand, the Order thinks that any Liberty vessels in the Omicrons are a violation of their sovereignty. This logic may be arrogant and/or insane, but this is FL, so anything less is a bit of a disappointment. They will shoot you if no other factors apply. Since other factors always apply, though, here are your words to live by in Omicron Delta- "alliance of convenience".
4) Whatever you feel like, but the Omicrons have a -lot- of caps, which is no small part of why it's an absolute blast to play in.
5) See 1. Even if they would probably be fine with it, asking makes them more amicable, which makes them less likely to leave you to die.
EDIT: btw, chances are you don't know where Dabadoru is, if you even know it exists. The Core are not the bounty hunters of house space... they don't give a damn that you're legitimate authority. In their eyes, "legitimate authority" is "he who has more firepower", and Liberty does not exactly have the force to threaten Core holdings in the Omicrons.