7 hour layover in Oslo
 

Can I reasonably take the train in (the one that is a few minutes longer than the airport-only train but far better value given that we are a family of 4) and visit the Opera House and the Viking Ship Museum? If so, how would I do this economically?
Prompt reply appreciated as we are leaving very soon!
Thanks!

Yes, but a 7 hour layover would probably give you 4 hours in the city allowing for exiting and travel time, then travel time and security etc. back at airport.
Buy a 24 hr travel pass for zones 4N to 1 at the public transport desk in landslide hall just after customs - cost NOK 190 for adults, ½ price for children, but free for them on Saturday and Sunday - will cover local train return to city and transport in city on train, tram, bus and metro, but unfortunately not the ferry to Bygdøy for Viking ships - take bus instead. Opera house is next to station.
Note that although the local train is only a few minutes longer in journey time it does not run as often - up to 40 mins between trains. You need to plan return journey!
https://ruter.no/en/
http://www.visitoslo.com

Thank you! Will definitely check times. Do you recommend anywhere for lunch/snacks near either of those places. Nothing formal, given the timing...

There is a cafe/restaurant in Opera and a cafe in the Folk museum next to Viking ships. Given your limited time you might put up with slightly higher prices there. There is a shopping centre in the same building as the train station, you'll also find cafes there. Norwegians aren't big on eating lunch out (most work places have a lunch room or canteen) - so not too easy to find reasonable hot food - but if that is a must then head for Grønlandsleiret - https://www.google.no/maps/@59.9131331,10.760443,17z