Would both life as well as elementary particles no longer exist upon “heat death? Would the elementary particles in the standard model still exist upon “heat death”?
Would electrons stop orbiting around the nucleus? Would the periodic table of elements still be relevant assuming every nucleus collapsed as a result of lack of energy so there aren’t any elements anymore? Everything is one messy separated bunch? 
 A: 
The heat death of the universe, also known as the Big Chill or Big Freeze, is a conjecture on the ultimate fate of the universe, which suggests the universe would evolve to a state of no thermodynamic free energy and would therefore be unable to sustain processes that increase entropy

You ask:

Would the elementary particles in the standard model still exist upon “heat death”? 

Heat death is a thermodynamic model, two levels removed from the quantum level where the elementary particles are defined. With heat death, interactions between elementary particles will stop because there would be no energy to interact with each other at the thermodynamic frame. But stable elementary particles may exist( see below)

Would electrons stop orbiting around the nucleus?

Not on the way to heat death, this is the quantum frame and it exists because no energy is gained or lost. When proton decay lifetimes catch up with the time of the universe, nuclei will stop existing so the electrons would become free.
The nuclei will collapse  when proton decay lifetime catches up with the timeline  of the universe, not because of lack of energy. 

Gravitationally, the universe will be dominated by dark matter, electrons, and positrons 

Proton decay takes care of the disappearance of quarks.   
You ask:

Would the periodic table of elements still be relevant assuming every nucleus collapsed as a result of lack of energy so there aren’t any elements anymore.

see above, only photons and electron positrons and neutrinos antineutrinos  are expected to exist after the heat death. 
Your title:

Would both life as well as elementary particles no longer exist upon “heat death?

Certainly life depends on the thermodynamic frame and it will long stop existing. 
It mostly  depends on the dynamical assumptions of the specific models, if neutrinos annihilate with antineutrinos and positrons with electrons before the final death , only photons will remain .
