I'm using the same entity type for multiple friendship queries, so fields are populated by context. It might help if you look at the downloadable source code, which has examples. Here's an example of FreindshipType.Show:
var friendship =
(from friend in twitterCtx.Friendship
where friend.Type == FriendshipType.Show &&
friend.SourceScreenName == "Linq2Tweeter" &&
friend.TargetScreenName == "JoeMayo"
"\nJoeMayo follows LinqToTweeter: " +
"\nLinqToTweeter follows JoeMayo: " +
Notice that I'm looking at FollowedBy properties of SourceRelationship/TargetRelationship properties.
FriendshipType.Exists is deprecated in Twitter API v1.1, which is the reason for the 404 Not Found. I've just removed it from LINQ to Twitter. Thanks for letting me know.