First, your oil ring is worn well past the service limit of 0.039" and would certainly cause the engine to blow oil smoke, which is blue. Note that either white or black smoke would have a different cause. Your replacement ring set should have two thin steel rail oil rings with a ring expander. Do not accept a ring set like your old cast iron one, which wears out too quickly due to not being chrome plated on the outer surface of the rings. Ensure that the top (compression) ring is also chrome plated, as the original one was. Put the first and second rings in the right way up (there will be a mark on the rings, and it faces upward). Do not remove the piston from the connecting rod, or there is a risk you'll put it back on the wrong way around.
There would not normally be a carbon build-up in that engine, but since it is burning oil, that would account for it.
The cylinder should not require honing, though it is good practice to break the glaze in the cast iron liner when replacing piston rings. I normally do that with some 400 grit abrasive paper or cloth, applied to the bore with my finger tips, using a circular motion with a diameter of about an inch. Unless the bore is damaged (visible vertical lines in its surface), I believe the use of hones usually does more harm than good. Carefully clean the bore after breaking the glaze, and ensure no trace of abrasive is left behind anywhere in the engine.
Your engine was running with grossly excessive valve lash on both valves - perhaps it has been run for years without adjustment, but if not, someone has made a mess of it. Bear this in mind when sourcing future servicing, in case you are employing someone unreliable. The correct clearances are intake 0.004", exhaust 0.006".
The positive crankcase ventilation valve for the engine is underneath the flywheel in a sealed compartment. In the compartment is a brown (phenolic-impregnated cloth) disk, which is the valve. Inspect the disk, which will show some wear on the upper side where it strikes the compartment's steel cover once every revolution. If it is deeply indented by wear, replace the disk. Check that there is no build-up of crud in the port underneath it. There is also a folded piece of plastic mesh (not steel) in a vertical compartment within the PCV compartment. Remove and wash the mesh, and while it is out, carefully push a 1/16" drill bit back and forth through the drain hole in the bottom of the vertical compartment, at the opposite end to the port where the breather tube attaches on the outside of the engine. If that tiny drain hole blocks the PCV compartment will flood with oil, which will then go through the breather tube into the air cleaner and from there through the intake port into the engine, where it will burn and possibly cause additional smoke.
You should service the carburetor while you have the engine apart. I suggest you watch this video, which shows the correct way to do this for your engine:
Bear in mind that many owners make a mess of the carburetor cleaning operation and get poor results afterward. Also, on older small Honda engines like yours, the black fibre gaskets on both sides of the carburetor insulator (between the carburetor and the intake port) always split when you dismantle the engine. They cannot be reused when this happens: use new gaskets on both sides of the insulator.
Please post some pictures of your engine as you go, and ask as many questions as necessary.
