I’ll be taking the Great Northern train from London’s King’s Cross station to Welwyn Garden City. Can I use an Oyster card for this? Is there a weekly pass that I can use? I’ll be traveling this route all weekdays for 2 weeks.
TFL have a map at https://tfl.gov.uk/maps/track/national-rail which shows where Oyster cards can be used. Welwyn garden city falls outside of this zone. Although there have been verious commitments and proposals to extend the scheme, like: http://www.whtimes.co.uk/news/oyster-card-setback-for-welwyn-hatfield-and-potters-bar-1-5267576
You can buy a season ticket valid for 1 week between any 2 stations, you can see the fare at: http://ojp.nationalrail.co.uk/service/seasonticket/search note that if you qualify for a railcard (https://www.railcard.co.uk) (you will likely quality for the Network railcard based on your question, and possibly more). It may be cheaper to buy the railcard and individual tickets rather then a weekly pass, this is because railcards offer a third off fares, but this does not extend to season tickets.
You can also purchase season ticks which include tube journeys, these may or may not be a cheaper option depending on your exact journeys.