If you are a seasoned County employee and decide to take a new position and that position gets cut, do you have bumping rights to your prior position?

Bumping rules are covered by each specific MOU, so you should check the relevant MOU.  But generally when a filled position is eliminated, the person in that position has bumping rights if she/he has more seniority than other people in that classification.  There are a lot of caveats to this, so check the MOU and call HR for advice.