On problem 1:
A. If we have 2 rows and n columns (2xn board) and we put a domino vertically in the upper right corner, then we have the rest of the board equivalent to a 2x(n-1) board. If we put the domino in the corner horizontally, we have to put another horizontal one below it, leaving the rest of the board equivalent to a 2x(n-2) board. This should give you the recurrence relation.
B. Consider n=1 and n=2 cases
C. Use your formula (creating a table may be fastest).