If AWOS reports IFR conditions but I can see at least three miles, can I take off or land?
In the specific case that you mentioned (VFR at an uncontrolled field), you can take off or land because the only legal requirement is to maintain VFR cloud clearance and it's your responsibility at the pilot to do that. The possible downside would be that if you have some kind of accident or incident and the FAA or NTSB investigates, they will take the AWOS reports as the reference for weather conditions at the time. If weather or visibility was relevant to the accident then your decision might be questioned, unless of course there are witnesses who can confirm the actual conditions.

In the case of a controlled field and VFR, ATC will base their actions on the reported conditions so they would tell you that the field is IFR only. One 'workaround' that could be useful is an SVFR clearance, which allows you to operate VFR in an airport surface area even below VFR minimums. You have to request it because ATC can't offer it although they may hint strongly at it to help you out: "the field is IFR, is there anything you'd like to request?" But ATC may not issue an SVFR clearance if the reported visibility is below 1 mile, so in the scenario you mentioned it wouldn't help you. (And note that you need an instrument rating for SVFR at night.)

Under IFR the picture is different. A part 91 IFR flight can legally depart in zero visibility so there would be no issue there, provided that you get your clearance on the ground first, before entering IMC. Legal in this case obviously isn't the same thing as safe, though.

I deliberately didn't answer your second question because I think it would be best for you to submit it as a separate question.