Can a Double 12 Domino set function as two Double 6 domino sets? If I subtract 6 from any domino with pips more than 6, do I effectively create two double 6 sets from a single double 12 set?

No, because you have too many dominoes. Also you won't even have the same number of copies of everything. You will get four copies of most dominoes, three copies of doubles other than 0:0, one copy of 0:0 and two copies of any other domino with a 0 on.

You can get two complete double-six sets by removing all dominoes with no 0 and exactly one side greater than 6, except for 6:12, then subtracting 6 from numbers greater than 6, but treating 6:12 as 0:0.