While taxiing, you're supposed to have the taxiway map up (https://flightaware.com/resources/airport/JFK/APD/AIRPORT+DI...), and something like ForeFlight (which many pilots, GA and commercial, use) will alert you when you're approaching a runway as well as once you enter it. Not sure what the instruments in a 20+ year old 777 will have to help with this (likely the taxiway map, unsure if they'll alert on runway entrance).
On communication: most modern flight computers do support getting your IFR clearance from the FAA (see CPDLC: https://en.wikipedia.org/wiki/Controller%E2%80%93pilot_data_... and Tower Data Link System: https://www.faa.gov/air_traffic/flight_info/aeronav/acf/medi...). This doesn't include taxi clearance.
Looks like ForeFlight and MITRE are working on bridging the gap (take the audio taxi clearance and overlay it on the map): https://www.mitre.org/news-insights/impact-story/runway-safe...
EDIT: here's a video of the feature in action: https://www.youtube.com/watch?v=9p5-7jVg8fE; looks like it's only on the most expensive plan though :/