Just to clarify, in case you don't know, what everyone means by cross the cables under the down tube is to run the shift cables and housing from the brifters around the front of the head tube to the cable stops on the opposite side of the frame from which they start and then allow the bare cables to cross back to their correct/respective sides under the down tube to route properly through the bottom bracket cable guide to the derailleurs.

Since the front brake and its cable/housing turns with the bar/fork, run your cables and housing so that your shift cables are not encumbered by the front brake cables...if that means the shift cable is routed to the inside then so be it....if that means it is routed around the front brake then so be it. This is more a matter of function over form.

I would leave the barrel adjuster in the frame stop unless the caliper arm is hitting it (unlikely). The fit and pressure of just a feruled brake housing may damage the threads in the stop negatively affecting future use.

-j