You are correct on #1, except that it is impossible to tell what A is because since B is a knave, we have no idea what A actually said, and therefore we cannot hope to evaluate whether it was true or false.
Explanation: We know because of what C said ("B is lying") that B and C cannot be either both knights or both knaves (This is the case regardless of what B actually said; if B said something true, and C said "B is lying", then B is a knight and C is a knave; if B said something false and C said "B is lying", then B is a knave and C is a knight.) Therefore, if A actually DID say "There is only one knight among us", there are two possibilities: First, A is a knight. Therefore he made a true statement, and B and C must both be knaves to comply with his statement. This contradicts our knowledge that B and C must be different types, so it can't be the case. The other possibility is that A and one of B/C are knaves, and the other is a knight. If this is the case, though, then A made a true statement (since there is only one knight), which contradicts our assumption that A is a knave. Going back, we must conclude that A could not have said "There is only one knight among us". This means B lied about what A said. Therefore B is the knave and C must be the knight.
You are also correct on #2.
Explanation: What gets a lot of people, apparently, is the notion of "inclusive" vs. "Exclusive" or statements. In order for the statement "Either X or Y" to be true, X can be true, Y can be true, or both X and Y can be true. The only way the statement is false is if both X and Y are false.
So if A was a knave (and therefore lying), both "I am a knave" and "B is a knight" must be false. But obviously "I am a knave" would not be a false statement if A were a knave. Therefore A must be a knight. And since the "I am a knave" portion of his statement was false, the other portion must be true in order to make the statement as a whole true, and so B is also a knight.
New problems:
1)
Suppose A says, "I am a knave, but B isn't."
What are A and B?
2) ("of the same type" = either both knights, or both knaves.)
A says "B is a knave."
B says "A and C are of the same type."
What is C?