Caching pages for anonymous users prevents other cache logic?
The page cache just takes whatever is generated by the current page, stuffs it into a large string and stores it into the cache. As long as that cache entry is then valid, it is returned and no other code is invoked.

Once the page cache is expired, the page will be regenerated completely and any other element will either be loaded from a specific cache or rebuilt.

If it's important that certain information is kept up to date, all you need to do is make sure that the cache expiration time of the page cache is the same or lower as the one of the relevant block/page/whatever.

There is a concept, called ESI, that basically allows to define placeholders in a page cache string that can be replaced with separate request. However, Drupal 7 doesn't really allow that yet (Drupal 8 will hopefully improve this) but this is for things that actually are different based on the user, for example if you have a site that is the same for all users except a single block that is user-specific. Not for something like a weather block that is the same for all users and valid for 5 minutes.