Either the tub should have been set in a layer of mortar or it should have been filled with water or at least two fat workers before the nails were set.
The problem is that the weight of the water has pulled the tub down slightly. Properly supported that wouldn't happen.
No, you should never put holes in the rim of the tub, it's guaranteed to leak at some point at the very least and it could very possibly crack the tub.
What to do now? If you can gain access to the bottom of the tub you could have low expanding foam put in there to fill the gap from the bottom of the tub to the floor. That would keep the tub from pulling down. Then fill the tub with water and caulk the seam between the tub and the tile with a good quality bathroom caulk. Allow the caulk to cure for a day or so before draining the tub.
There is certainly no reason to pull the tub and put in a shower.