How are intermolecular forces between non-metallic substances affected when the substance is heated to its melting point?
First of all, multiple intermolecular forces could be present between molecules, and, yes, all forces influence attraction, binding and etc. However, if you study any chemical system, you will see that is usually dynamic.
It all depends on conditions like temperature, pH and especially compound properties, etc. For example, the drug captopril (an antihypertensive drug) could form multiple bonds with the protein angiotensin-converting enzyme (ACE).:

How about one specific atom of a molecule? In theory, one atom could form multiple intermolecular forces at a specific time (let's say picosecond), however, it is not practical - energetically unfavorable. Imagine that atom should be in such a position, an electronic configuration that reduces its potential energy and also form multiple intermolecular bonds. It is possible that such systems exist, but they are not common.
Second answer:
Yes, intermolecular forces have specific values of bonding energy:

For example, it takes less energy to break London dispersion forces than to break hydrogen bonds.
Third answer:
It all depends on the level you want to know... Basically, when you heat molecules (give them more energy) they absorb it and move faster.  Vigorous movements make bonds break. For example, water molecules at 100 °C start to move so fast that almost all its intermolecular forces are broken. Fun fact: you can smell lemons at room temperature because it has volatile molecules (molecules that form very few intermolecular bonds) called Limonene.