Is the whole world's airspace covered by ATC?
The entire world is divided into Flight Information Regions (FIRs), so in theory there is someone responsible for everywhere.  However, those FIRs are then subdivided into different classes of airspace.

Class A/B/C/D/E airspace is "controlled", meaning there should always be some sort of ATC available.  In remote areas, though, ATC may not have radar coverage, so separation would be based only on position reports (plus ADS-B, for participating aircraft).

Class F/G airspace is "uncontrolled"; any ATC service would be advisory only, and around untowered airports, they'll tell you to use CTAF.  Often the reason airspace is uncontrolled in the first place is a lack of radio and/or radar coverage.