Answer to Question 1
First, you would accurately measure a given volume of the liquid and then weigh this volume. Dividing the weight by the volume used gives the weight per unit volume, which is the density in the units used for the measurements.
Answer to Question 2
In a commercial melting point instrument, the melting point capillary rests on top of a metal block (usually aluminum) that is being heated at a constant rate. Therefore, during the time it takes to visually recognize complete melting of the solid, the temperature is observed as a range, Tinitial to Tfinal, not a point.
Answer to Question 3
No, this was not a wise decision. Some organic compounds decompose upon heating. Therefore, the compound present at the end of the melting point determination may not be the same as the original sample. Rerunning the same sample can lead to erroneous data.