It’s been a long time since I taught students, but I found that having 3 or 4 areas to cover each lesson was helpful.
* Tone production - embouchure, breathing, intonation, possibly dynamics. This is done with long tones, though today I would emphasize playing with a drone tone.
* Technique - scales, arpeggios, tonguing, legato.
* Repertoire - work on a specific tune or composition or etude.
* Improvisation - ways to deal with improvising on the current tune, usually one or two specific approaches like using a pentatonic, or a certain pattern.
The improv area was for jazz students, clarinet kids didn’t get that part. The last two areas often included a listening component, especially for jazz. Focus on feel and specific performance techniques, like articulation, scoops,etc.
I always played along, even on the scale and long tone parts. I would listen to the student play alone, then demonstrate how I did it, then we would play together, then the student alone again. In jazz tunes we would play the head together, and if the student could improvise, trade 4s. I comped badly on piano, nowadays I would use a backing track.
I think twice a week is fine, especially for the first 6 months. After that, once a week is best. Regarding the fee, I would find out what the going rate is for your area and charge this guy 1/2. He is paying for your experience, and you are giving him a discount because this is a new area for you.