Rather than renting the equipment fios provided for my TV and internet, canI purchase it from FIOS and save the monthly rental fee
You could. But then you would need to return the rental and get a receipt. You could also run into an issue where Verizon would think the bought router was still theirs and that it was just swapped out. You can find this issue via google search.
but yes you can purchase a new Gateway Router if you desire.
You can only purchase the router.
But you can not purchase STBs for TV service.
Your only option for that is to purchase a cable card device like TIVO.
You then have to pay a rental on cable card.
And Cable card devices can not access on demand content.