Is it really necessary to properly heat oil in a pan before frying veggies?

If you are looking to fry veggies then yes, it is necessary to pre-heat. If you put them in a cold pan with oil, you start 'sweating' the veggies instead of frying them.

For example: if you put a sliced onion into a pan with hot oil, it will cook and have a nice golden brown color as it caramelizes. Put that same sliced onion in a cold pan with cold oil and then add heat, and the onion will first turn translucent and lose moisture.

Both of these cooking methods have their uses, but they are different.
How do you know when the oil is ready to cook? It will start to shimmer a little before it reaches the smoking point. Put the veggies in when you see this shimmer, or at the very first wisp of smoke.

You heat up the oil first because your food tastes/feels less greasy oil is hot (around 350 degrees F) than when it is cold (under 300 degrees F). So if you want your food to resemble the greasy food of your iconic dive-y restaurant there is, I guess, no need to heat your oil. Otherwise, do heat it. You usually want it hot enough that if you toss a few drops of water in there, they sizzle.

As to how high to turn up the burner when heating, usually most recipes that I've worked with suggest medium-high for many applications, except perhaps searing steak. When working with olive oil specifically you don't want the temperature to go too high because olive oil has a low smoke point. Depending on how refined the olive oil, the smoke point can be as low as 374 degrees F (reference). You don't want to heat oils past their smoke point, because they begin to break down at that point. In order to avoid doing so, I wouldn't heat at 100%.

To sauté, the dry pan should be brought to a medium/medium-high temperature. At the correct temperature, if you drop a spoonful of water into the pan, it should form a single solid mercury-like ball that glides across the surface. Once the pan is heated, the oil can be added. The oil will come to temperature in a trivially short period of time.

To heat the pan, it's best to set the heat at the point you want it, and allow the pan to come to that temperature, instead of setting the heat very high, as this will risk overheating the pan.

For sautéing, use an oil with a high smoke-point, I prefer grapeseed oil because it retains its health benefits even when cooked at a high temperature, unlike say, olive oil.

My chef busts of me for this but in my opinion it’s actually more efficient... I will have a stack of sauté pans in the back left burner of my station and I keep the burner on low. Say I need to sauté asparagus, I’ll grab the lowest pan which is ripping hot with my tongs, add the asparagus and then drizzle oil over it. The oil shoots up in temperature quickly enough to start frying immediately, then I’ll add my garlic to sauté immediately after and then hit them with veg stock. You don’t have to wait for your oil to get hot and then drop veg and splash hot oil or burn yourself. But your pan has to be super hot in order to do this properly. It’s faster and easier.

First answer is yes, it matters, sometimes. Food reacts differently to different temperatures, particularly when you're aiming for certain coloring. The type of metal pan you're using also reacts differently to oil and heat so as to stick or not stick. The most important reason you keep your oil hot is in a restaurant environment, where you don't want to waste time cooking when you can simply keep your oil hot (spend a few minutes upfront, when you're not busy) and start cooking instantly.

For most home cooks however, this doesn't matter.

The second answer is it depends. Different oils have different smoke points. http://en.wikipedia.org/wiki/Smoke_point