Can you mix cards from the CCG and LCG? Do they count towards tournament play if you do?

You can mix cards from the CCG (black-border) with those from the LCG (white-border). However, you must use opaque sleeves because the card backs are different between the two editions (CCG cards have "Call of Cthulhu: Collectible Card Game" written on the back, whereas LCG cards are blank).

You can't use CCG-era cards in tournament play unless it's a non-official tournament and the organizer allows them, or in the case of an officially sanctioned tournament, they have been reprinted in white-border format. In the latter case, you must notify your tournament organizer of any black-border cards included in your deck.

Personally, my group uses them all the time. There are some very neat black-border cards, and the expanded card pool allows for theme decks not possible using only white-border cards.