If you are growing them on a trellis, make sure to put the fruits in slings/hammocks made of nylon netting produce bags or nylon stockings/tights as soon as they are baseball/orange to softball/grapefruit size. You need to secure the sling in such a way that the weight stress is taken off the vines and fruit stem, and the fruits have room to grow. You will likely need to adjust the sling as the fruit grows -- check weekly.
tried to nibble the stem, I imagine the fruit could have fallen. Was the break clean? Jagged? Long strip of skin pulled off on one side? (You need to think like a detective and look for evidence
The red sap is most likely natural sap of the vine. Here's a photo of a drop of sap welling out of a muskmelon vine where I pinched/cut off the end (...what can I say? I thought it looked kind of pretty