CDNs are typically used for transferring large objects, for whom the anycast routing instability is a real concern. If a client's anycast endpoint changes in the middle of a connect, the client will receive an immediate RST from the new server.
Major CDNs do use anycast routing, in conjunction with DNS/geolocation based routing in locations where anycast's results would be unpredictable (usually in locations where there are many POPs.)
In theory yes, but BGP anycast routing is really bad in performance terms. BGP doesn't care about performance, only politics and cost, so you can get all sorts of weirdness.