No, your turn does not end
Part 1: Full Attacks & Grab
A perfect example for this is the Dire Tiger, which has 3 attacks, all of which have the Grab ability.
So you're a dire tiger, and you make a full attack. You hit with your bite, choose to grab, and succeed on your check. You now have the Grappled condition, which states:
A grappled creature takes a –2 penalty on all attack rolls and combat maneuver checks, except those made to grapple or escape a grapple.
It does not say anywhere in the Grab special ability that your turn immediately ends. Additionally, in the Grapple section it says:
You can take any action that doesn’t require two hands to perform, such as cast a spell or make an attack or full attack with a light or one-handed weapon against any creature within your reach
Specifically, this is referring to the creature being grappled, because the grappler is unable to take a full attack having used their standard action to maintain the grapple. However, in the case of Grab no standard action was used, and the grappler has the same Grappled condition that the grapplee has.
Because there is nothing in Grapple or Grab that specifically prevents a grappler from making a full attack (aside from a standard action that Grab ignores), and it is specifically stated that someone with the Grappled condition can take a full attack, you as a Dire Tiger may continue to take attacks after your Grab at a -2 penalty.
Part 2: Greater Grapple
Yes, if you have not used your move action for the turn, you can immediately roll to damage the grappled creature as a move action.
Once you have grappled a creature, maintaining the grapple is a move action. This feat allows you to make two grapple checks each round (to move, harm, or pin your opponent)
Part 3: Rake
You're still a dire tiger, and you've remembered you have the Rake ability.
A monster with the rake ability gains two free claw attacks that it can use only against a grappled foe
It does not say you rake based on a maintain check, it only says you get the claw attacks to use against the grappled foe. No, you do not get to rake twice in the same round (you can only use the two claw attacks once).
Edit: Additionally, with Greater Grapple and Rake you still cannot use Rake on the first turn, because Rake states:
A monster with the rake ability must begin its turn already grappling to use its rake—it can’t begin a grapple and rake in the same turn.
Part 4: Constrict
If you were instead an Octopus with 8 grab attacks and constrict, you could take advantage of this line:
You can, as the creature that initiated the grapple, release the grapple as a free action
This would allow you to Grab, Constrict, release, and repeat for each grab attack you possess. However, keep in mind:
Free actions don’t take any time at all, though there may be limits to the number of free actions you can perform in a turn.
Your GM could still limit how many times you can release the grapple.
Edit: Again with Greater Grapple, Constrict states:
A creature with this special attack can crush an opponent, dealing bludgeoning damage, when it makes a successful grapple check
If you chose to hold the grapple until the next round (and possessed Greater Grapple) you could make two grapple checks and do extra constrict damage with both.
Best Answer
Yes, you can initiate a grapple and pin in the same round.
That is exactly what Greater Grapple is for, it explicitly allows you to make an additional check as a move action -- and the wording of grapple says nothing about having to wait until the next round to use any of the special grapple actions.
If you're worried about the word maintain, that seems to just be a poor choice of words on the designer's part. Greater Grapple specifically states that you may make such checks twice in one round; and the rules for maintain make it clear that this is the only type of offensive check that you make in a grapple. The various actions such as pinning/moving/etc only occur on a successful check to maintain: