Will line conditioner do anything for the other receptacle in a duplex?

No I agree with Jason, although I have not tested it. However there are a number of wall wart devices that you plug into a duplex receptacle which are supposed to help reduce noise everywhere, or in the adjacent plug. Based on reviews I have a few of these around the house. I haven’t done any detailed testing. At some point I found it just isn’t worth testing personally.

If that is the case then at some point you might want to try some things that actually work. Because I have them, and there is no "testing" you plug em in and hear the improvement immediately.

It will depend on its design. Ask the manufacturer. 

It should, as the these pieces are designed to take a "dirty" incoming signal, and send out a "clean" one.  The "clean" signal it sends out goes to your gear *AND* back to the outlet/duplex, hence a "clean" duplex (theoretically a whole circuit).  The effect will *not* be as strong on the duplex (or circuit), but you should get a little bump.Cheers!

@millercarbon 
me too!