When do I need to worry about a course deviation indicator reverse sensing?
On this approach plate, which was talked about here, why is the back course on two different sides (one is shaded on the left, one is shaded on the right)?


The 108.5 IPKN localizer is there only to mark safe path out for missed approach procedure and is intended to be flown outbound, so it has the signal swapped so it indicates normally when flown outbound.


  How does a front course and a back course affect your CDI settings?


It does not affect settings of CDI at all, because ILS does not take the OBS course into account. It indicates left when it senses more of the left signal and right if it senses more of the right signal, independent of anything else.

What it affects is:


The left signal is on the left when flying the normal direction and this applies both before and after the transmitter. So when you fly in opposite direction, the left signal is on the right from your point of view, causing the reverse sensing.
With HSI you can compensate for it by selecting the normal course in the OBS. The instrument then still senses the same, but the instrument is now upside down, negating the reverse sensing.



  Are there any difference between these two CDIs?


No, since these are both CDIs.

There is a difference if you have HSI:



The difference between CDI and HSI is that on CDI the needle always moves left-right, indicating whether your position is left or right of the selected radial (via OBS for VOR, but fixed for ILS) independent on direction you are flying. So if you are actually flying the opposite direction, you must steer the other way.

In contrast the inner part of HSI rotates with compass, always pointing the arrow in the selected direction. This has two advantages:


you see how your direction of flight relates to the selected heading and
when you are flying in the opposite direction (backcourse), the needle is upside-down, negating the reverse sensing.



  And also, why is it labeled a VOR receiver?


Because it is (also) used to display VOR signal. Both CDI (where the needle always moves left-right) and HSI (where the inner part with the needle rotates with compass) are VOR indicators.

Note that almost always the same tuner is used to receive VOR and ILS signals and the same instrument is used to display them though the processing logic is different (VOR detects phase while ILS detects differences in the depth of modulation from the two signals).