Request context
Any request that hits a web farm gets served by an available IIS server. Context gets created there and the whole request gets served by the same server. So context shouldn't be a problem. A request is a stateless execution pipeline so it doesn't need to share data with other servers in any way shape or form. It will be served from the beginning to the end by the same machine.
User information is read from a cookie and processed by the server that serves the request. It depends then if you cache complete user object somewhere.
Session
If you use TempData
dictionary you should be aware that it's stored inside Session
dictionary. In a server farm that means you should use other means that InProc sessions, because they're not shared between IIS servers across the farm. You should configure other session managers that either use a DB or others (State server etc.).
Cache
When it comes to cache it's a different story. To make it as efficient as possible cache should as well be served. By default it's not. But looking at cache it barely means that when there's no cache it should be read and stored in cache. So if a particular server farm server doesn't have some cache object it would create it. In time all of them would cache some shared publicly used data.
Or... You could use libraries like memcached (as you mentioned it) and take advantage of shared cache. There are several examples on the net how to use it. This is just one.
But these solutions all bring additional overhead of several things (like network and third process processing and data fetching etc.) if nothing else. So default cache is the fastest and if you explicitly need shared cache then decide for one. Don't share cache unless really necessary.