Algorithm to build tri-nodes in MapleStory
Problems (a),(b),(c) can be solved in polynomial time; simply enumerate all possible ways of choosing a subset of 6 cards and test each subset to see whether it satisfies all of the constraints.
If you care about a practical algorithm, I would suggest using a SAT solver.  It is easy to express this problem as an instance of SAT.  Use boolean variable $x_i$ to represent whether the $i$th card in the deck is chosen or not.  Then you can write boolean constraints that express each of your constraints.  I suggest using an off-the-shelf solver like Z3 to solve this instance of SAT; they will tell you whether any solution exists, and if so, give you one example.  If you want to find all solutions, look up algorithms to find all satisfying instances to SAT.
Problems (d),(e) are more challenging but can also be solved in the same way.  Iterate over $m=5,4,3,\dots$ and for each $m$ use a SAT solver to test whether there exists a solution for that value of $m$.  There are at most 120 different cards, so you can imagine having a deck and a sidebar of all the other cards not in the deck; let $x_i$ represent whether the $i$th card in the deck is chosen, and $y_i$ represent whether the $i$th card in the sidebar is chosen, and express as an instance of SAT.