After studying the rules at length, I came to an entirely different conclusion about the mechanics of grab/grapple/rake, so I'll throw in my 2 cents. I'll refer to the attacker as "you" since you are playing as a druid with wild shape, but these same mechanics would also apply to monsters with grab/rake abilities.
1) When does rake activate, is it a free action to take the rake action immediately upon the successful check to maintain the grapple or do I have to do another action first?
Rake activates in two circumstances: while pouncing (if you have both pounce and rake attacks), and as a free action against a grappled foe when you begin your turn with the grappled condition. I think everyone agrees on the former, but there is some confusion regarding the latter. In my opinion, it's a free action taken at the beginning of your turn which is separate from the maintain grapple standard action. It must be performed against a grappled foe, which means that you can only rake the foe you began your turn grappled with. Finally, it does not occur on the same turn that you initiate the grapple (with the exception of pounce, which seems to be a special case).
2) If it activates during one of the grapple actions (damage, move, pin, tie up) does it have to be the damage action or does something like moving trigger it too?
Starting your turn with the grapple condition triggers it, NOT the maintain grapple standard action. If you go on to successfully maintain your grapple, then you would still choose a grapple action as usual.
3) Does the rake ability do automatic damage when using it or is it an attack(s)?
It is an attack, so you still need to roll for it.
4) If it is an attack do the grapple penalties apply on the attack roll?
Yes, but as others pointed out, the grappled opponent also has a dex penalty that balances this.
Okay, think that's it for rake. Now for Grab:
5) If I successfully grapple someone on the first attack of a full-round attack, can I continue to make all of my attacks for the round, with the penalties of the grappled condition?
Yes, you can. Also, I believe the grapple condition penalties apply to both you and your opponent- UNLESS you opt to "hold" your opponent with the grab action instead of grapple. To hold, you take a -20 penalty to your CMB check to initiate and maintain the hold. If successful, your opponent is grappled, but you are not. Since you do not have the grapple condition, you would also not have access to your rake attacks on your next turn. If you are able to maintain the hold on your next turn, you automatically deal the damage indicated for the attack that established the hold.
6) Upon successfully maintaining the grapple, does the grab ability automatically deal the natural attack's damage?
No. The automatic damage is for the HOLD, not a grapple.
7) Could I, with both rake and grab, deal: grab damage + rake damage + "damage action" damage?
No, there is no such thing as "grab damage" in and of itself. Automatic damage from a hold and rake attacks are mutually exclusive.