The clunky shifting may be partially, or completely, a matter of mastering an unconventional shift technique. These are semi-automatic clutches. That said, most folks don't really understand what that means. Fewer still understand, learn and master the shifting. At this stage of the process, I would first learn how to adjust the clutch preload, then get the feel of the shift lever. What actually happens when moving the lever is the same as pulling the hand lever, with a manual clutch. Holding the shifter at either extreme (up or down) releases the clutch and the engine freewheels...or it should, if the preload is properly set and everything inside the motor is correctly assembled. Practice modulating the throttle and shift lever movement/release. You should modulate the throttle the same as you would with a hand clutch. Generally speaking, you want to give the shifter a quick stab, then hold the lever, briefly, before releasing it. This is a little awkward to learn, at first. But, once mastered, you will be able to shift as smoothly as with any hand lever-actuated manual clutch.
I suggest sorting out the clutch & carburetor before taking the bike apart. The carb rebuild should be pretty straightforward, with only a couple of CT70-specific quirks. The fact that it runs as well as it does likely indicates an easy carb rebuild.
Of all the various components on this bike, the engine is the easiest to completely restore...relatively speaking. Any and every part that could be needed is available and most are cheap. If you are capable of tearing down the engine and reassembling it, then there's probably $300-500 worth of parts between you and a mechanically like-new motor that can easily deliver a 5-digit service life, with timely maintenance...mainly oil changes and periodic oil spinner cleanings.
If you're planning a total restoration, it really makes sense to go through the engine, i.e. splitting the cases for a thorough cleaning and inspection. I'd expect to find worn shift forks. That was a common problem when these bikes first reached the showrooms. Honda revised the shift fork design for much better service life. These things are easy to replace, once the cases are split. And, worn shift forks, plus pins, can hurt shift quality.
As for the photo upload issue, it's being worked on...