There are going to be a lot of rough patches in your pursuit of cello, particularly at the beginning.  The chaos you are experiencing does not seem abnormal to me.

As for "taking time from the fundamentals" in your private lesson, at the beginning, EVERYTHING you play is an opportunity to learn the fundamentals.  Your private teacher will know how to best use your lesson time.

Touching students has fallen out of favor in recent years because of a few cases of abuse.  However, I have never, ever taken lessons from anyone who didn't occasionally touch me--it's just easier sometimes than trying to explain it in words.  Unless it's abusive physical contact, I'd suggest you learn to tolerate it.

By being in a class that is a little too hard for you, you might be in a position to learn more quickly because of the additional pressure.  Most people really don't want to play with others who are below them in level, so maybe this is a blessing?  

As for the administrative SNAFUs, I have no real help.  There was obviously some miscommunication somewhere.  If you are this unhappy, ask for a refund.  I wouldn't worry about the consequences for the teacher.

Finally, you said you are learning how to count the notes you can't play--that is a PRICELESS skill.  The very first and most important thing is to be able to COUNT.  I cannot emphasize that enough.  The class may be worth the cost if all you do is learn to count.  Play whatever you can, and count silently for the rest.  When I was in high school and the music was way too hard for me, sometimes I just was happy to play the first note of every measure.  Hee.  

Have cello, will play Pachelbel.  Enthusiastically.