Should a post in a circular concrete form go through the bottom?
Concerete around a fence post is to help with lateral stability. It's not being used as a footer, since there isn't a whole lot of weight involved with a fence post. 

As such, there's no need for concrete to be below the bottom of the post...in fact, you like don't want that, as you want any water that gets between the concrete and wood to have a place to drain.

What I'd suggest doing is to use gravel at the bottom...tamp it down with the post, set the post vertical, then add a bit more gravel and tamp down to 'set' the post. Then you can apply concrete around that, above the ground and then slope the top away from the post to help shed water.

Depending on your soil type, you may not even need concrete--you could actually just use gravel all the way to the top, tamping as you go.