One post at a time, please. The ST90, like the CT90, needs a good battery for ease of starting and these bikes are fundamentally different in the ignition and electrical power departments than a CT70. What do you have for DC volts measured at the points spring? Should be more than six volts with a healthy battery and the run-stop switch on the handlebar set to run. If no power at the points spring, start backtracking up the points wire until you find it and repair the break there. Could also be a bad ignition switch.
The ST90 also shares the same shift drum stopper problems that the CT90 suffers. Most likely, the shift drum stopper has popped off the star wheel on the shift drum. Or the star wheel is loose on the drum. Easy fixes but hard to get to. You will need a JIS #3 screwdriver to get the clutch side case screws out. They only look like a Phillips head . . . trust me here, please. You will need a clutch nut socket after that and a clutch holder tool to hold the clutch while removing the clutch nut. A JIS #1 screwdriver is needed to remove the screws from the outer clutch cover. Again, please trust me here and buy some JIS screwdrivers or you will soon be drilling the heads off of a lot of screws.
You will likely need to clean the sludge out from under the outer clutch cover because it is a job that has probably never been done. There is also a black rectangular thing on the lower right front of the engine center case. Grab it with a pair of needle nose pliers and remove it. This is the engine oil screen and it should also be cleaned and re-installed.
Please let us know how the work is progressing.
Rick