Identifying the normal form that a certain relation complies with
All attributes are prime, they belong to at least one candidate key. 

Therefore, the relation is - trivially - in 3NF.

If you look at the requirements for both 2NF and 3NF, the exceptions start with "every non-prime attribute ...". Since the relation has 0 non-prime attributes, these requirements are trivially covered.

About your analysis, the first parts are OK (finding the candidate keys, that the relation is not in BCNF and that it is in 3NF).

The last part, about 2NF, shows confusion. Check again the 2NF definition and how do you define partial dependency and transitive dependency.