Burris Timberline 4.5-14x w/ Ballistic Plex reticle

[Sam Vimes]

Is it possible to range find with a 4 - 14 x 32 scope?

Depends on how you define rangefinding. If you mean rangefinding by PX then it will be possible but it'll be quite unreliable. This scope has nowhere near high enough mag or large enough objective to rangefind by PX reliably and at range. To do so would necessitate minimum magnification of around 30x and a front end of 40mm or more.

Is the depth of field the same on an AO scope as on a non AO scope, assuming they are both set to the same range. So if I set this scope to 30m and just leave it there will it be the same as a non AO scope set at 30m?.

AO should have little impact on the DOF of a scope. That's a product of the magnification and objective size. In this case, for a given level of mag, the 32mm objective should ensure a greater DOF than a scope with a larger front end.

Does the longer eye relief increase, decrease or have no effect on the chance of parallax error. It will not always be possible to focus the scope correctly?.

This depends entirely on who you talk to. However, picking a sensible PX distance and an appropriate mag setting should mean that it won't need the px reset for every shot. In fact this is precisely the scenario that HFT shooters find themselves in.

Will it be as bright as a cheap Hawk 40mm scope and suitable for lamping?.

Decent optics and the coatings of those optics play a large part in the light transmission (brightness) of a scope. In the case of the small objective Burris the greater quality of lens may well cancel out the extra 8mm of objective that the Hawke has (for a given mag setting).