What is the proper way to run ethernet cable along walls and doors
I have used those cord covers.  They are pretty easy to use and fairly stiff so they protect the cable.  The adhesive they come with sticks pretty well.  I'd try to keep them on the baseboards and moldings rather than the drywall so you can clean up any adhesive residue when you remove them.  

I don't think they have the fittings you need for corners, angles, etc.  They don't help at all if you have to cross a floor.  For your application you'd be just as well tucking them in by the baseboard as suggested.  

It's really unusual that you can't cover an apartment with one wireless router.  It may be as simple as repositioning the router up high on the wall.  I'd see if you can fix the wireless before bothering with the raceway and all.