Catalan numbers, paths on a grid I'm struggling to understand the concept of the idea. Some questions "push" me to use the reflection method which is hard for me to imagine and understand. I prefer to look at it as a sequence, but check out this question:
Consider a $7\times7$ grid whose points are $\lbrace (i, j) | 0 \leq i, j \leq 6\rbrace$ and a set $D = \lbrace(1,1), (2,2), (3,3), (4,4), (5,5)\rbrace$. How many paths of length $12$ that start from $(0,0)$ and end at $(6,6)$ do not cross the points of the set $D$?
My solution was $2^*C(5)$ using the reflections idea, which I'm unsure is even right. Is there a way to solve this kind question more abstractly? (with sequences for example)

Yes, your solution works. An alternate explanation is: Say your first step is moving to (1,0). Then you'll need to pass through (6,5), and moving from (1,0) to (6,5) without touching the diagonal is the same as moving from (0,0) to (5,5) without going below the diagonal, which can be done in $C_5$ ways. Of course we could also have started by moving to (0,1) which gives another $C_5$ so the final answer is $2 C_5$.
I'm not sure how your proof works because you haven't explained it much. I'm guessing you're referring to this reflection argument? Anyway if you want feedback on your proof and not just your answer then you'll probably need to give a link to the reflection proof you're looking at + explain how you're using reflections in this particular problem.
