Q: If I purchase a T10 ticket, am I able to use it to go from the airport to the Franca train Station for one of the ten journeys? Or, is there an additional fee? I understand that I need to switch trains at Passeig de Gràcia.
A: You can use the T10 ticket for your journey from Barcelona airport to Franca Station and it will only count as a single journey.
As you pointed out change at Passeig de Gràcia station and then take the yellow line metro to Barceloneta. You then have a short walk to Franca station. The Airport and Franca station are all within the Zone1 area and provided your journey takes less than 75 minutes in total (which it should do) you will only use one journey on your T10 ticket to go from the airport to Franca station.