Inside the engine compartment the air is hot from passage through the radiator and the air-conditioning condenser, plus some heat from the engine, and dirty because it's next to the engine. In fact, with modern concerns over emission control, there's not really much coming off the engine that isn't already ducted away elsewhere (crankcase emissions, fuel vapour and so on) so "dirty" is probably the wrong word to use, so you're right to point that out.
If you have the undertray on, then there shouldn't be any road dirt getting in to the engine compartment. But the whole compartment is designed to be a low pressure area (to enhance airflow through the radiator) so it's "dirty" airflow, you lose the ram effect. However, outside of competition cars, this is really not worth a lot.
I haven't seen any proper test data comparing the various induction modifications: most of what we see in forums is anecdotal, people's opinions, often influenced by the fact they've spent money on a modification and that the sound is different. I've seen some data that shows most filters have very little significant difference in flow restriction: enough to make a difference in competition (where you're at full power almost all the time, and a few HP can make the difference between winning and losing) but not significant for daily driving.
I suspect that the best benefits you can give your engine in terms of induction changes are taking the PCV through a catch can to a separate filter, removing it as a source of contaminated air into the inlet, and deleting EGR (for those of us that have it, which is all bar turbo-petrol engines, I think). These two emission control systems put more hot and dirty air into the inlet than even the worst inlet/filter system could ever be responsible for.