I assume that is the habit of using Daedric Alphabet while they've already got used to dealing with the Daedra. Take a look at old Fyr and how he and Aryon used Daedric Alphabet for the safe sake.
In everyday life and texts with no need to secure, common alphabet is more likely to come.
Those were dunmer. Dunmer are very different from Altmer.
It is almost certainly similar to the Falmer, Ayleid, and the two Dwemer fonts (the Skyrim and Morrowind fonts vary a bit). I'd make it more stylized and complex, similar to how the Falmer alphabet is, perhaps drawing inspiration from http://3.bp.blogspot.com/-E_68BRRtQcI/T778owP7YmI/AAAAAAAACgc/eKdXA5aHDl0/s1600/islamic_calligraphy9.gif http://upload.wikimedia.org/wikipedia/commons/8/80/Muhaqqaq_script.gif.
I thought the Dwemer were culturally isolated from the other Mer. They never left Tamriel, and all that jazz.
Otherwise, I'd go and look at the samples we have of Ehlnofey, and maybe even the Dragon Tongue, and add some influences from there. The Altmer are all about keeping ancient traditions pure, and as such their language would probably preserve many elements from the ancient languages.
As for the script, that's really up to you. I personally think it should probably be more like Ayleid than Falmer. One real-world influence you might use that I've always thought would be nice to see would be http://en.wikipedia.org/wiki/Hangul#Morpho-syllabic_blocks which would fit in with the Altmeri obsession with order.