graduate admissions - Asking for Recommendation Letters for Multiple Grad School Applications

I would like to apply for many math PhD programs in order to maximize my chance of success.

My main question is how should I ask for recommendation letters. In particular, should I ask for a generic letter (in the sense that it's addressed to "the admission committee") to use for all my applications? Or should I ask for multiple letters from the same person for every university I want to apply to (presumably the letters have the same content, except maybe in who it is addressed to)?

Also, most PhD application deadlines are in December. But I finished undergrad at the end of last year. Should I ask for these letters now so as to not increase the time between when the professors know me and when they write the letter (this may mean I'm sending a letter dated February in November), or should I ask closer to the deadline around November this year?