I was wondering if there was a case anyone knows of that I can transfer the parts over from my old case, but would fit my new requirements, which are 6.5" in height and 16.5" in depth.
I was looking at a couple cases at NewEgg, here
The two cases you linked should be fine. They don't get much smaller than that without going to a Mini-ITX form-factor.
Also considering building a Micro-ATX build. Are the parts for a Micro-ATX case typically cheaper than an ATX build?
The only difference is the case and motherboard. All other components are standard size.
Cases tend to be cheaper because they're smaller, with fewer bays and expansion features, reducing the materials cost and construction complexity.
Motherboard prices are almost the same as Full-ATX, since other than the loss of a couple expansion ports, they still have most of the components of a Full-ATX board.
I think the price gap would be larger if Micro-ATX sold better. Full-ATX is still much more popular with the build-it-yourself market.