As your daughter is in mainstream schooling, I do not see any way that the clause regarding who she calls mother and father is relevant. Within six months of my youngest being in school (he was only a year old when his dad and I split up) he was calling his stepfather dad, because he so badly wanted to have someone to talk about when school talks about fathers- and it took a while before we picked up on it because school hadn't realised this wasn't normal for him. Unfortunately, your ex is not the only influence in this area.
As far as meeting half-way goes- I have a big problem with this. I'd suggest that if you ask for this, your child support payments should be generous enough that it isn't a choice between your daughter travelling half-way to see you and the whole family eating that month. Again, this is the problem we had- but if your ex is not on a low income, it's less of a concern. (The situation- I was left in 13,000 pounds worth of debt, with no maintenance or CSA AND he wanted me to transport the children to him. It wasn't possible at the time.)
I'd define major medical event, and specify whether you want to be involved in discussions about treatment for long-term conditions- eg asthma, eczema.
I'd ask for arrangements to be made so that the school sends a copy of school reports, etc. directly to you, and specify that you wish to play a part in deciding which schools, etc, your daughter attends. This is going to require a time commitment on your part. I'd also specify that you must be listed as the second next-of-kin on all your daughter's school paperwork, though you're too far away to help in an emergency- it will stop the school dismissing you.
The Christmas thing- alternating Christmas has never worked for us. When the boys weren't with us for Christmas, they got bigger presents on their birthday, a single present at Yule and we partied at Hogmanay. When they weren't with their dad, Santa Claus visited his house twice in one year- which just seems childish to me. I'd seriously consider having another holiday in the year as being your extra-special family time with your daughter- basically, you need to do some soul-searching on this one, it's not as straightforward as it seems.
In addition to phone contact, I'd ask that your ex sets up an email account for your daughter. By writing and emailing, you can help her with literacy- plus children love letters
As the pp said, I'd establish now that your dd should have a passport, but that she should not go outside the country without your permission.
Other than that, it all looks pretty good.