If AWOS reports IFR conditions but I can see at least three miles, can I take off or land? The uncontrolled airfield where I have my airplane gets foggy in the morning occasionally, but there have been times when there is just a little fog where the AWOS station sits so it reports 1/4 mile visibility and undetermined ceiling, yet the runway farthest from it is clear and there are no clouds. My flight instructor and ground school instructor said you must always comply with the AWOS reported conditions, yet my flight examiner (I just got my private pilot license) said a pilot can overrule the automated station if it is clearly wrong. Machines can break and are limited to where their position is.
FAR 91.155 talks about visibility, but doesn't mention how it is determined. Flight visibility is determined by the pilot, so I interpret this to mean he can overrule the AWOS if it is clearly wrong.

(d) Except as provided in §91.157 of this part, no person may take off or land an aircraft, or enter the traffic pattern of an airport, under VFR, within the lateral boundaries of the surface areas of Class B, Class C, Class D, or Class E airspace designated for an airport—
(1) Unless ground visibility at that airport is at least 3 statute miles; or
(2) If ground visibility is not reported at that airport, unless flight visibility during landing or takeoff, or while operating in the traffic pattern is at least 3 statute miles.
(e) For the purpose of this section, an aircraft operating at the base altitude of a Class E airspace area is considered to be within the airspace directly below that area.

Another question about (e), after reading it about 5 times: my airport is class G up to 700 feet, then turns into class E above that. Does (e) mean that if I am at 700 feet AGL, I am still in class G? Of what significance is this?
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.