I'm sure someone with a closer match to you will respond soon. My dd didn't have a peanut allergy (dairy is our main problem), and she's not anaphalactic.
When my dd started all day K last year, I sat down with the principal, an office secretary (because she usually administers all the meds and epipens), and the district nurse for a 30 minute meeting. I also had a separate 30 minute meeting with the teacher.
A few days before the meeting, I emailed everyone a couple documents that I wrote. One was a list of scenarios that I brainstormed might be a problem for my child (i.e. in your case things like lunchtime seating, snack time and touching classmates, lessons involving food, birthdays). Then for each scenario, I wrote down what I wanted the teacher/school to do. (i.e. for birthdays, I asked for a list of birthdays for all children in the class so that I could know when to send a special snack for my dd) I also made a document listing my child's allergies, what her reactions are like, and what measures we take at home when we have the reactions. (For us, most of our problems were avoided by making it clear to the school that our dd was not allowed to eat or drink anything that was not sent from home.)
As far as peanut free table or snacks, I think would go in there and ask them what experience they have had in dealing with peanut allergies.
Then I would explain how severe or different your child might be compared with others. You have to tell them about your child's allergies. The office will have no idea exactly what no peanuts mean because every child is different. So you will have to tell them about the degree of severity of the allergy. Child should not eat peanuts/Child should not touch peanuts/Child should not touch someone who has touched peanuts/Child should not breathe peanuts/Child should not be in the same room as peanuts/Child cannot have plain M&Ms. You also want to tell them how severe the reactions are because schools encounter a whole range varying from life-threatening and immediate, to delayed and annoying, to painful, etc. You also want to tell them if you have ever had to use an EpiPen and/or ended up in the Emergency Room, because that will make them understand how serious the issue is. (Not sure about this one, but you might want to bring an expired epiPen and an orange to quickly demonstrate its use if they don't know how to use it.) Does the EpiPen need to be in the classroom? Because it will most likely reside in the office. Does the epiPen need to travel with the child's class from one extra to another (like to gym class)? These are things that you need to decide are either necessary, or overkill, depending on your child.
Then I would ask them what kind of accommodations they have made for allergies in the past, and use that as a starting point.
I would go in assuming that they will be willing to do what you need, so I would suggest that you walk in with a specific plan, and be ready for the school to brainstorm with you about how make some minor tweaks so that it will work for the school. (i.e. My dd has a problem with even touching pumpkin innards, so both the principal and the teacher asked that I be present for the pumpkin farm field trip.) I did have the impression that they were glad that I walked in with a list of specific measures that I wanted them to take, so that they didn't have to guess. Remember, at this point, you are the only person in the room who understands the severity of your child's allergies and what does and doesn't work for your child. Their job is to let you know if what you need will work with the dynamics of the classroom, and if it doesn't, help you brainstorm a way to make it work.
I have read of one child who had an airborne peanut allergy whose plan was that she would bring five of her own clean place mats each week for lunch so that she had a clean surface to put on the table each day at lunch. Then she had no children sitting directly across her or next to her, but diagonally from her was fine. I don't know if that would work for you, but I think the school should be able to help you come up with something that would work.
Yes, you will have to have a conversation about snacks in the classroom. Will it be important for the other children to thoroughly wash their hands? I have heard of parents asking for a peanut-free class? (My dd's principal wanted to send a letter to the parents of my dd's classroom alerting them about the allergy. I declined, but maybe you want to write such a letter to be sent home to the parents and include words like "die" and "life-threatening" where appropriate?)
So I hope that if you walk in and say from the start that you all want to figure out how to make your child's learning experience effective, and at the same time find a way to accommodate your child's needs without too much extra work from the teacher, that the school will be glad to do what you need. I found that, except for one incident, I was more than happy with the way my child's school and teacher dealt with our needs.
Oh yes, an acquaintance of mine has a child with special needs, so she sets up meetings like this all the time to discuss her child's issues (not allergy, but meetings that regularly take up extra time from the school staff). She suggests bringing food. She really goes all out by bringing lunch. In my case, I brought a pie, and it was appreciated.