Does the number of each type of electron orbital remain the same for each energy level?

I think what your asking is, is that does each type of orbital exist as you move across and down the periodic table, eg. you move from each energy shell.

As you go through electron shells, you add another type of orbital.

it goes:

1s

2s

2p

3s

3p

4s

3d

4p

5s

4d

etc. etc. you usually don't go farther than the 4p shell.

So take Phosphorus

it has an electron configuration of 1s(2)2s(2)2p(6)3s(2)3p(3)

* the number in parentheses is the number of electrons occupying the shells

I hope this makes sense. Basically once you reach a 3p orbital and move farther through the energy shells, a 3p orbital will always exist.

Yes. A p orbital, whether it's a 1p, 2p,3p..... will always have 3 orbitals.