"Is it advisable to run a new 10 gauge red from pos. of starter batt.
to one end of cut-off relay (switch portion) for aux batt., as opposed
to using the stock 12 gauge red to supply 12V to same?"

I think people generally prefer to run a wire directly from the positive
pole of the starter battery to the aux battery.  The rationale is that
the "stock" wire probably has a longer run length and may go through
several connectors before supplying current to the aux battery.  This
additional length and extra connectors translates to lower voltage at the
aux battery than at the starter battery.  Lower voltage limits how well
your aux battery get charged.  By running a beefy wire directly from the
starter battry to the aux battery you deliver a good voltage to the aux
battery.  The size of the wire you use depends on how much current you
want to be able to deliver to the aux battery.  There is a post in the
archives where somebody measured a deeply discharged Optima deep cycle
pulling 50amps.  I don't think you are going to see much more than that.
There are tables on the net that display the relationship between wire
gauge, run length, and the max current that the wire can handle.  You
should double check this, but I am pretty sure that a short run of 10gauge
wire can handle over 50amps.  This is what I used in my van.

BTW I think the "can" is a solenoid.  Check out the solenoid that GoWesty
supplies for aux batteries (http://tinyurl.com/3carma).  Does it look like
the little can in the picture?