How does FAA type certification relate to intellectual property
rights?

Simple answer is that it doesn't .... not one little bit.

For example: everything in an O-235 has been out of patent for
years now. Same with old aircraft designs. Is there any reason why
somebody couldn't tool up and start ripping out O-235s, and selling
them new?

No not really ... HOWEVER you would have to prove to the FAA that you have
all the drawings and all the engineering data to support such an activity.
There may also be tooling involved. Possession of this data is really the
critical issue. There are examples, one fairly well known involving a
current production airplane, which I won't name specifically here, in which
such data was obtained by the backdoor and the airplane put into production
be persons who apparently have little right to do so. It is for this reason
that OEM's so closely guard their data. If you chose to do this you would
also have to jump through the hoops to get a production certificate.

There are other options though to produce parts covered by existing TC's
without such data or 'ownership' of the type certificate such as a PMA. See
FAR Part 21 for the details of this. There is also an FAA Order 8130-2C
which will answer most of your questions. Get these documents from the FAA
web site.

If you have anymore specific questions let me know.