You can! Just make sure the size of the foam cage works with the opening of the vase. If it’s too small you may use a small lomey dish (clear plastic tray that comes in different sizes).
It really depends. I usually try to do foam pieces no more than 24 hours from the even because it’s not a direct replacement for water. Sometimes flowers do great in cooler temps and sometimes they dry out. It depends a lot on the season and region you are in, and types of flowers/greenery used. Lots of variables to think about.