Variation on partial Set Cover with penalties
This answers question (2):

The greedy heuristic for Set Cover / Maximum Coverage always picks the set which contains the maximal number of uncovered elements.

Assuming your modification for the heuristic is picking the set which greedily increases the solution profit, you might end up in a lousy approximation ratio.

Consider the following example:
$$A = \{a_1,a_2,\ldots a_{k}\}, B=\{b_1,b_2,\ldots b_k\}$$
$$C = \{C_1\} \cup C_2$$
$$C_1= A\cup B \setminus\{b_1\}$$
$$ C_2 = \{\{a_{i},b_1\}|i\in[k]\}$$

The greedy heuristic will output $C'=\{C_1\}$, so the profit is $1$, while the optimal solution is $C_2$ which gives profit $k-1$.