Yes, orbitals do overlap, a lot. The "outer" orbitals extend farther, that's all - keeping their fuzzy limit in mind.
For instance, all s orbitals have their maximum density (per volume unit, not per radius unit) right at the nucleus.
They are orthogonal in the sense that the sum of their product over space is zero, which implies that they change their sign at different places.
So for carbon:
- 1s is + everywhere;
- 2s is + at the center and - farther away;
- one 2p has a + lobe in one direction and a - lobe in the opposite direction, hence is orthogonal (integral of product) to all spherical orbitals;
- an other 2p has + and - lobes too but in a direction perpendicular to the previous orbital, so the integral of the product is zero with the previous too.
Note that any linear combination of the three 2p orbitals is an orbital too (because these have exactly the same energy) and that other choices are just as good to make bases for the 2p orbitals. The two-lobed set is convenient for chemistry and bonds but is just one possible choice: the one with a zero orbital momentum around the lobes axis. If you add with 90° phase the two previous 2p aligned on x and y, you get the orbital with the definite angular momentum of 1 around z, which looks like a doughnut instead of a peacock. Three doughnuts make a basis too, and for instance their limear combinations make the peacocks again. Other linear combinations would make elliptic orbitals, which are stable and have just no definite angular momentum around any axis. A 2p electron has no reason to follow a peacock nor doughnut shape.
If you come from an optics or radiocomms background, the 2p base is the same story as the linear and circular polarization of the EM field. Better: the shape and direction of a 2p corresponds to the polarization of a photon emitted during a transition to 1s.
Higher orbitals make more complicated combinations as there are more of them in a basis.
One should remember too that the known orbitals are only for one lone hydrogen atom. We use them at other atoms because they're understandable, not because they're accurate. Repulsion among electrons deform all orbitals because they overlap so much, and this changes the energy a lot. It also explains why individual electrons use to occupy all the accessible orbitals (2px and 2py for atomic carbon) instead of pairing on a single one. The same happens with O
2, and in a more complicated form, in transitions metals.
Nice representations of orbitals there:
http://winter.group.shef.ac.uk/orbitron/