The simple answer is no, not every approach procedure is lined up with the centerline of the runway. For those approach procedures that are not lined up with the centerline the visibility requirements and sometimes ceiling requirements for the approach are higher to allow the pilot room to manuever. Basically Bryan, the higher the visibility requirement the farther out you will see the runway giving you more time to line up with the centerline.
To answer your secondary questions.
It is certainly to your benefit to be lined up with the centerline of the runway when you touch down... after all, they are much longer than they are wide.
We do not necessarily align the heading bug with the runway centerline. A heading merely points the aircraft in a certain direction. If a crosswind is blowing the aircraft will drift left or right of centerline if only flying runway heading. So, the pilots adjust their heading to "track" the final approach course whether a VOR, NDB, SDF, LDA, GPS, RNAV, or ILS (or any other arbitrary sequence of letters you can dream up) then also track runway centerline during the final stages of approach.
I hope this helps.