Q: If I purchase a T-Casual 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.
Important update: On 01 January 2020 the T10 ticket transport ticket was discontinued and replaced by the T-Casual ticket. Click here to learn more.
A: You can use the T-Casual 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 T-Casual ticket to go from the airport to Franca station.
A better alternative to the T-Casual might be the Hola Barcelona Travel Card that offers unlimited travel on Barcelona public transport. You won't have to worry about the number of journeys you have left on your T-Casual Ticket or about purchasing a new one that you might not entirely use. With the Hola Barcelona you can travel unlimitedly and worry-free while saving money.
Hola Barcelona Travel Pass