After seminary are priests allowed to go to grad school say to get a Phd or STB (Bachelor of Sacred Theology)? If so are you able to obtain both of these degrees as a priest?
Yes. I know priest who have both, especially religious order priests. Just look up the CV of a few Bishops and you’ll notice they have both.
Ordination to the priesthood does not preclude any further education. However, it does usually mean that such further education is at the prerogative of their bishop. They are, of course, under obedience once ordained.
I’ve known plenty of priests who have gone on to earn a number of different degrees after ordination (not all even directly related to theology). I’ve even known a few who probably wouldn’t have chosen to pursue such degrees if they weren’t “encouraged” to do so by their bishops.
After all, every diocese needs priests who have expertise in canon law, liturgy, etc. More often than not, that type of specialized further education happens after the priest has already gone through seminary and been ordained (since the seminary requirements are usually pretty standard for all the priests).
I’ve encountered lots of priests with advanced degrees. It’s not obvious because a priest with a PhD. degree is still called Father and not Doctor as he would be outside the Church.