You asked pertinent questions.
For R44 systems descriptions start with the POH, free online at
https://robinsonheli.com >Publications >R44 Series (in dropdown menu) >R44 POH (2nd dropdown menu) thence Section 7 - Systems Description
Further info can be had in Powerplant chapters of the Maintenance Manual and Ilustrated Parts Catalog, if desired.
The R44's engine cooling air inlets on the aft cowl were designed to capture MR (main rotor) downwash when hovering to minimize hot exhaust gas ingestion to the fanwheel inlet.
Fanwheel-supplied air also cools:
1. Muffler (also used as source of cabin heat)
2. Drive belts
3. MGB (main gearbox)
4. hydraulic reservoir (includes pressure regulator)
5. cylinders
6. oil cooler(s, two on fuel-injected version)
7. alternator
8. magnetos
9. fuel flow divider (fuel-injected version)
10. battery (when mounted in engine compartment)