I have the old toilet removed for plumbing repair and since it's old and
dirty, I want to replace it with a new one.

A few questions:

There are one-piece and two-piece toilets. One-piece looks easier to clean
(no gap for water to hide). Is there any disadvantage, other than high
price? Are there some brand that's better or worse?

Every floor flange photo I found on the web shows the flange mounted on top
of the floor. But mine is mounted flush with the floor:
http://www.smugmug.com/gallery/12478717_2d2zy

Is this a problem? Shoud I use a thicker wax ring, or a flange extender?