My little lime tree has flowered with over 20 little limes; however, the limes are falling off shortly after the petals fall from the flower. What do I need to do...this is happening to all of them even the larger ones. I have noticed some small white bugs...I think they are mealy bugs and I have hand picked off all that I can find...the tree is oozing sap in some areas on the leaves...cleaned this off with water. No ants at all on this tree...it is now indoors under a grow light. Any help would be appreciated.

Did you look for scale insect? If you didn't see scale before it's hard to notice.
The dropping of the fruits it may happened as a response to the move from outside to inside or because it doesn't get as much heat/light as it needs to support them.

I would spray your tree. I had a similar problem but my bugs looked like some kind of brown egg sack that really turned out to be some kind of bug stuck to the tree. my tree is also dripping sap at the leaves