Is it a good idea to have the dentist pull out my child's loose baby teeth, or just do it myself?

Usually not needed.. Usually not needed. Rarely, if a tooth is in the way of the child's comfort or ability to chew, and is not loosening further, the dentist can remove it. This may be faster and cleaner than a parent's fingers.
It depends. In the vast majority of cases, baby teeth just fall out normally on their own and should be just left alone. Some children start working on loosening it themselves just to get it out faster while others avoid doing anything to the tooth and wait way past when it could be easily removed. If it needs to be nudged out, it is best left up to the dentist, who will gladly take a look and advise you.
Loose baby teeth... ...Usually fall out by themselves. If they resist a gentle pull, they are not ready to go yet.