First relax and take a deep breath. Kindergarten is a huge adjustment for just about everyone and with moving out of state to a school district that you are unfamiliar with and a crazy schedule you've got more than your fair share on your plate.
My DD5 is in full-day five day a week kindergarten. The first two to three weeks really was all about adjustment: working on separation anxiety, lining up, getting around the building, eating in the cafeteria, walking to specials (20 minutes of a 40 minute class slot was spent getting to and from, it's a large school and this has since improved) and figuring out each child's capabilities. (Students were individually administered the GKAP, GA's kindergarten exam and evaluated. This took a full two weeks for 21 kids.) Since your DD has only had 13 days of school she is still technically in this adjustment period. With them going to school only every other day, I suspect the adjustment will take longer. Kids with separation anxiety must have a horrible time with this schedule. I know my DD would.
Here's a couple of things I would do before trying to make any changes. If you haven't already, you need to do your homework. By this I mean, get a copy of your school's kindergarten curriculum. Once you've read through it, decide if it meets your expectations and your daughter's needs. If it doesn't ignore the rest of my post and find alternative education for your DD.

If it does, watch your DD's class closely. Is the teacher covering the curriculum? (Ours is listed monthly, so it's pretty easy to follow what is being covered.) If she is not and is just coasting through the day waiting for the dismissal bell, that would be a reason to speak to the principal and or teacher. If she is, then you will probably have to suffer through her teaching style for the year. Given your every other day schedule, I would give this to about Thanksgiving break to determine how the classroom is functioning.
I would continue to volunteer, though I would probably not make classroom management suggestions. While in the school building, I would try and casually observe the other sections of K. Do they take as long to line up? Are they walking quietly in the halls? On your way to DD's class, peek in another class or two and see if they're sitting quietly or if their classrooms are chaotic as well. Is a behavior management plan being followed?
Get to know the school and the parents of the children in DD's class. The room parents for the K classes at my DD's school have set up playdates at neighborhood parks for each K class, so that the children have some time to play and the parents get a chance to chat and compare notes. Contact your room parents and see if they are willing to set this up. Or, get a copy of the class list and arrange a group playdate yourself. You can even approach it as you're new to the area and would really like to meet more parents in the neighborhood. We do ours on weekends or after school, but with your schedule you could even do this on your alternating days.
I understand where you are coming from with your disappointment in your DD's Kindergarten experience. My DD5 got the
one teacher out of ten sections of K that I did not want her to have. Her teacher is similar to yours in that she has 25+ years teaching at the school and is set in her ways. She also has a reputation that precedes her and most parents dread having their child in this class. Unlike yours, she is extremely strict. She doesn't bend, she snaps and she is easily frustrated with her large class size and focuses her attention mostly on behavior. I thought my DD would be miserable in such an environment. But you know what, she has adjusted and she loves it! I, on the other hand still have reservations, but I'm dealing with them.

I volunteered to be class parent representative and it really has given me an insight into this teacher and how she deals with her class and though I may not always agree with her I am learning why she does things the way she does.
Take this advice with a grain of salt. Good luck!
Mary
PS: Our school has been in session for 9 weeks already.