Yes, technically you can use the same network dataset for car driving and biking as long as you prevent travel on the LineVariantElements and have some reasonable way to accurately model travel time and (particularly for cars) various restrictions for travel. For bikes, depending on what you want to do, you might want to allow travel on transit lines that can accommodate bikes, which is fully supported by the transit data model.
Unfortunately, although you an model driving to park-and-ride, and you can model riding transit from the park-and-ride to the destination, you can't model both of those within the same analysis. This is because the Network Analyst solvers don't have a way to track an internal "state", which in this case would be a flag like "passenger currently has vehicle with them". The passenger would start their journey with the vehicle (flag=True), then park at the park-and-ride and get on the train or bus (flag=False). They get off the train or bus, and since flag=False, they have to walk the remainder of the way to their destination. There's no way to model that because the Network Analyst solvers just aren't built to do that. So, you have to break up the analysis into two pieces. First, model travel to the park-and-ride by car. Then, model travel from the park-and-ride to the destination by transit and walking. Finally, combine the results.
For your last question, it's probably best to model your GTFS for different years in separate networks to avoid confusion. In particular, if you're doing an analysis for a generic weekday like Wednesday, and both GTFS years have recurring service for Wednesdays (specified in calendars.txt and not calendar_dates.txt), you could have problems with both services seeming to be available at the same time. I think you could do it if you were careful to always use specific dates that fell within the date range of only one, but this seems overly complicated and risky. If you want to save yourself the trouble of manually creating several network datasets that all share the same configuration, you can create one manually and then use Create Template From Network Dataset and Create Network Dataset From Template to automatically create additional ones using your other data.
