This is the perfect itinerary for a 10 day stop on the Faroe Islands. When traveling to the Faroe Islands I recommend you to rent a car as you can cover more locations.
You need to rent a car once you are here on the Faroe Islands as it is easier and you get faster from one location to another… There are many car rentals here but I recommend 62N car rental. https://www.62n.fo/carrental/en The reason is this company has a hotel as well and if you buy a package from them you will get a better price for the rental car.
My opinion is that you stay 2 - 3 days on Vágar island and I recommend the following places where you can stay. Hotel Vágar is literally just 5 minutes walk away from the airport. https://hotelvagar.fo/en/
If you want to be on your own I recommend to stay at The view cabins that are located at the village Bøur, just 10 minutes drive from the airport. https://www.theview.fo/
Feel free to contact me if you have any questions about the itineraries. I am here to help :)