I suspect something is not right with the stereo installation.  It may be a
wire laid in the wrong place that gets pinched and shorts when you shift or
it could be that the installer tapped into the wrong wire or wires while
connecting the system.  Of course, he could have bumped a connection
somewhere, too.

I agree that a stereo installation should not cause any of these problems
but not all installers use the same rule book.  Since it started happening
after the installation, I'd start by going beck to the shop and asking them
to check it over.  There should be no charge for them to do this.  Don't
forget to mention the inside lights thing, too.  You might want to make a
list of symptoms to take with you.

Let us know what happens, we can all put our thinking caps on together after
you check this part out.

PoD