There really is too much information missing to really answer this question, but in order of best performance I would actually skip the floor and go with a sealed crawl space. The second best would be closed cell foam followed by batts with a layer of foam board attached under the joists.
You definitely need to make sure any water issues are dealt with, the space dried out and the ground is covered properly.
Exactly what would be best in your situation would have to be diagnosed by an experienced energy auditor or contractor specializing in crawl spaces.
In answer to your questions above - no that would not encourage mice to move in, but your crawl space should also be sealed up properly to prevent them from getting in.
As for the plumbing, no as it would be covered in foam which is not really an issue as the insulation will help protect it in all but the most cold conditions and will last for many, many years if installed properly. Unless you are moving items in a kitchen or bath, the accessible parts needed are above the flooring.
For more information:
Read "What is the best type of insulation to put into our crawlspace to keep the floor warm in the winter?" a Q&A answered by Michael Holcomb.