You missed (a,f), and also in the matrix.the ordered pairs are:
(a,a), (a,c), (a,d), (a,e), (b,b), (b,d), (b,e), (b,f), (c,c), (c,e), (c,f), (d,d), (d,f), (e,e), (f,f)
For (b), try finding the supremum of a and b.
For (c), the dimension is at least 2 because otherwise the order would have been linear. It is indeed possible to find a realizer consisting of two total orders. This concept is new to me, and I don't know any theorems about it, so I used some trial-and-error. One of the orders I have is a < c < b < e < d < f.