What happens to energy changes in non-reversible processes?
$\Delta U = q + w$ and $dU = dq + dw$

This is the first law of thermodynamics.  It is not an inequality.

$dU\leq TdS-PdV$ is correct when there is only PV work. It is incorrect when there is other work such as electrical or gravitational.


  the right hand side represents the work done on the system by the surroundings. Please clarify if that is correct.


The PdV term alone represents the reversible work, the TdS term represents reversible heat


  Secondly, at a constant $S$ and $V$, spontaneous changes occur only when $dU \leq 0$. What's going on here? Is it as simple as: the system must lose internal energy in a spontaneous change or does it mean that work must be done on/by the system in a different way (eg. electrical work)?


the system must lose internal energy in a spontaneous change