There are others more qualified to answer these, but I'll take a crack at it. Everybody let me know if I'm remembering any of this wrong. First, here's a pic:
How thick is it?
Very thin. Like Pete said, .001" kind of scale.
Is the thickness a constant or does it vary?
Grows nonlinearly with the distance from the leading edge.
Does thickness vary with speed?
Speed affects the boundary layer profile, as velocity is one component used to solve the Navier-Stokes equations, which define the theoretical profile of a boundary layer.
Does it vary with turbulence?
Turbulence is basically rapid changes in free-stream velocity, so as the velocity changed, the boundary layer would as well. Standard boundary layer theory is a static analysis of airflow, though, and I don't know of any theories on dynamically changing velocity. Probably doesn't amount to much practical difference, since the effects we generally care about would just relate to the average flow velocity, anyway.
Does the boundary layer vary in thickness over the surface of the airplane in flight due to temperature or altitude?
Density of the freestream flow affects the boundary layer, and temp/altitude would affect that, so I would say probably yes, but not much.
Is the boundary layer thickness different at different points on the airplane in steady state flight?
Yes, reference Navier-Stokes and picture.
Does the boundary layer ever separate from any surface of an airplane in flight?
A severe separation of the boundary layer on a wing is what we commonly refer to as a "stall". Anywhere that the flow across the surface reverses direction in relation to the freestream is considered a separation, and results in a lot of drag (pressure drag). Un-faired wheels hanging in the wind will have separated flow behind them. Getting most of that flow to stay attached is basically what gives us the 10+ knots of speed with wheel fairings. Flow will often separate where there is an abrupt decrease in cross-section. I'm unsure if flow separates behind the bubble canopies on RVs, but is another place that flow separation is often fought.
Does the thickness of the boundary layer add to the physical cross section of the airplane and thus add to the drag?
Yes, that's basically what gives us skin friction drag. The way I understand it, laminar flow has less drag basically because it's smaller, therefore less cross section and less drag.
If it truly has zero velocity with respect to the surface of the airplane then it does not flow with respect to the airplane so how does laminar flow or turbulent flow of the surrounding air affect it?
The air around the airplane, outside the boundary layer, is assumed (reasonably) to be constant and parallel. That's not perfectly true, of course, but the variations usually don't significantly affect the outcome or analysis.
Since the boundary layer is not moving with respect to the aircraft surface but the aircraft is moving through the atmosphere there must be a dynamic interface between the boundary layer and the air that the airplane is passing through - how is this interface defined?
Basically the interface IS the boundary layer. Remember only the FIRST molecule of air touching the skin is at zero knots. The boundary layer is the explanation of how the first layer of molecules are going 0 kts in relation to the skin, the next layer is moving slightly, and each layer continues to go faster until some layer reaches freestream velocity (defining the edge of the boundary layer).
If the airplane strikes and object in flight what is the effect on the boundary layer?
The flow would be disturbed for a moment, but then re-establish itself immediately with no noticeable effect.
In oil drop experiments to determine airflow where is the oil with respect to the boundary layer?
The oil drop would basically be a wall in front of the flowing air in the boundary layer. The interaction that boundary layer flow with the oil is what tells you how the air is behaving at that point. Smooth laminar flow would just cause the oil to streak along with it. Turbulent flow (being turbulent, by definition) would hit the oil with more random velocities, causing the oil to splotch around a bit as it streaked (more turbulent = more randomness and "splotchiness). If the oil streaks backwards, against the flow, and is extremely spread around - that would be a separated boundary layer.
Whew.
