Will electrons pair up in an orbital only when all orbitals in different sub-levels have one electron?

Usually, yes, but not necessarily.

For a given #n#, different sublevels have differing #l#, where #l = 0, 1, 2, . . . , n-1#.

Hund's rule states that in order to maximize spin multiplicity, an atom usually fills one orbital at a time, and pairs up after every orbital of the same energy (or neighborhood of that energy) is half-filled.

Well, a counterexample is that thorium (#"Th"#) has a #7s# orbital that is pretty much the same energy as its #6d# AND #5f# orbitals (empty circle vs. empty triangle vs. filled diamond):

And yet, its electron configuration is:

#color(blue)([Rn]7s^2 6d^2 5f^0)#

rather than... (as lower #n# is usually lower in energy)

#[Rn]7s^2 6d^0 5f^2#
(Aufbau)

or... (as it decreases electron pairing interactions)

#[Rn]7s^1 6d^3 5f^0#
(Hund)

or even more seemingly reasonable... (as lower #n# is usually lower in energy)

#[Rn]7s^1 6d^0 5f^3#
(Aufbau + Hund)

And the reasoning behind that is the #5f# orbitals are particularly compact compared to the #6d# that #"Th"# would rather have #[Rn] 7s^1 6d^2# rather than #[Rn] 7s^1 5f^2#, even though (or rather, BECAUSE) they are all pretty much the same energy.

The #6d# orbitals are also apparently not so diffuse that a #[Rn] 7s^1 6d^3# ends up being viable, so that the #7s# being doubly-occupied leads to the lowest-energy ground-state thorium atom.