Pre-EDIT: As I go to post this, I see that a better and quite well-constructed answer by @rddr has already been posted (above, no doubt) while I was writing. Nonetheless I will leave this here along with the links below as the work is already done. Among the links is a heat transfer calculator that allows you to input the thermal conductivity as well as the thickness and surface area, so you can experiment with different sized and shaped pans.
You’re on the right track with thermal conductivity. In fact, thermal conductivity is probably the best answer to your title question as stated. If I am interpreting correctly what you’re trying to get at, another property you my want to consider in trying to think this all through is specific heat.
As you mentioned, material can transfer heat energy in three ways, conduction, convection and radiation.
What you’re describing in the stem is of course many processes occurring over the same time frame. They are essentially all additive, meaning if you could calculate each one and put them all together (with simple addition and subtraction), it would then be fairly straightforward to determine the temperature of the oil at a given time. However, the math/physics to determine the values of all of those components, and as well as how the processes interact with each would be fairly complex. If you want a real life answer for your kitchen, the easiest way will definitely be to test it empirically. If you want to do the math, or at least see the math, I can give you a few links to all the formulas and constants. But keep in mind this is a dynamic environment you’ve described, and it’s therefore not as simple as plugging in some numbers to the formulas. E.g. the thermal conductivity of a fluid such as air is dependent on its pressure and temperature. . . so when the air near the hot pan or stove is heated, its thermal conductivity also changes. Of course it’s also whisked away and quickly replaced with ambient air due to convection.
The Wikipedia page for this kind of thing I typically very good for jumping around and exploring related topics:
Specific heat capacity
For the next one, scroll all the way down past the table of thermal conductivities to find a brief summary as well as a calculator that lets you test different thicknesses and surface areas:
Conductive Heat Transfer Calculator
k = thermal conductivity (W/mK, Btu/(hr ft °F))
s = wall thickness (m, ft)
A = surface area (m2, ft2)
dT = t1 - t2 = temperature difference (oC, oF)
Solids, Liquids and Gases - Thermal Conductivities
This last page includes an Online Air Thermal Conductivity Calculator, where are you can plug in different temperatures and pressures, as well as some nice graphs:
Air - Thermal Conductivity vs. Temperature and Pressure