However, I never use the cleaning kit in the butt trap of my Garand to clean my rifle. I just have it to complete the rifle.
If I were you, I would get a commercial one-piece cleaning rod in a smaller caliber that will work in multiple rifles. 3 piece cleaning rods are convenient because they're easily stored away... BUT I can't tell you how many of them I've snapped off in my rifles.
I use a tetra cleaning rod in .22... Its small enough fit my 6.5 Carcano and Arisaka and strong enough to push patches and brushes thru my .30cal and 8mm bores. Plus, it has a protective coating that won't scratch the barrel.
In terms of other cleaning supplies, I use a brass bore brush soaked in Hoppes solvent to loosen up the fouling, then run Hoppes soaked patches down the barrel followed by dry patches until they start coming out white instead of black. I finish it all off with a patch with a few drops of gun oil on it.
I use dental picks and q-tips to remove fouling in hard to reach places...
For lube I suggest a grease instead of liquid... I use tetra grease. A local gentleman I know who I would consider an expert uses Mobil 1 grease on his match Garands.
I wouldn't use anything to clean the stock beyond a damp wash cloth to remove surface dust and grime.