What makes oil stick to the pan so bad that it is so difficult to wash out

As you suggest, there are a number of factors, but the best rule of thumb I've found for pan-frying is to use just enough oil to cover the bottom of the pan by swirling it around. In general, you don't want puddles (you'll get splashed with hot oil), but if the pan isn't covered then your food might not get cooked evenly.

These days I mostly just eyeball it, and err on the generous side. Using slightly too much isn't likely to hurt your dish - unused oil just won't get absorbed - but using too little will result in under-cooked or inconsistently-cooked food and probably a lot of gunk stuck to the bottom of your pan. So if you're not sure, just add a little more than you think you absolutely need.

The typical rule of thumb is that if it's a non-stick pan you do add a little oil to the pan first before heating. Most manufacturers usually recommend this to extend the life of the non-stick coating.

For regular pans (those without non-stick coating) you should heat them dry until you can feel the heat radiating from the surface when your hand is held about 6-inches above the bottom. Add your oil at this point. You'll actually need to use less oil because the same amount will spread across a greater surface area due to its decreased viscosity as it heats. Plus, your oil will heat up instantly and when you add your food it's less inclined to stick. Most people get impatient waiting for pans to heat (and in general) and this also ensures that the food isn't going into a pan with oil that's cold or not hot enough. When cold oil goes into a pan and cold food ends up on top of it you'll end up with one big sticky mess. As for adding oil before heating the pan, the longer fats heat without anything else in the pan, the quicker they'll break down and burn.