No, a successful grapple check or attack with the Grab ability does not end your turn.
To address your examples:
#1: A creature with a bite attack that has the grab ability, but also has 2 claws without grab.
If the bite hits and the CMB check is successful, both the creature and its target gain the grappled condition. That condition states, in part:
A grappled creature takes a –2 penalty on all attack rolls and combat maneuver checks, except those made to grapple or escape a grapple. In addition, grappled creatures can take no action that requires two hands to perform.
Claw attacks don't require two hands, so you're good there. They also don't grapple or escape a grapple, so the creature will take a -2 penalty on its claw attacks, but it still gets to make them. It's worth noting that this penalty roughly offsets the -4 penalty a grappled creature takes to its Dexterity.
#2: Something with the Greater Grapple feat.
The Greater Grapple feat is a source of much confusion. It does partially allow you to make (up to) two grapple checks in a round, but only after the grapple has been established. To understand why, let's look at the overall flow of grappling, omitting grab attacks for now.
A summarized grapple flow looks something like this:
Attacker: As a standard action, makes a successful CMB check to grapple. Attacker and Defender both gain the grappled condition. No damage or other effects happen at this point because the grapple was only just established.
Defender: Tries to escape in some manner but fails. They're really having a bad day.
Attacker: Must decide to either release the grapple (free action) or attempt to maintain it. As a standard action, makes a successful CMB check to maintain the grapple. This deals damage, results in a pin, or whatever other relevant nastiness the Attacker wants to do to the poor Defender. Attacker still has a move action to do something.
A few things change with Greater Grapple. As the feat states, the check to maintain the grapple becomes a move action instead of a standard action. This gives you two attempts per round to continue grappling your victim. The general flow now looks like this:
Attacker: As a standard action, makes a successful CMB check to grapple. Attacker and Defender both gain the grappled condition. No damage or other effects happen at this point because the grapple was only just established. Because the Attacker isn't maintaining a grapple, no further checks are possible at this time.
Defender: Tries to escape in some manner but fails. Still can't catch a break.
Attacker: Must decide to either release the grapple (free action) or attempt to maintain it. As a move action, makes a successful CMB check to maintain the grapple. This deals damage, results in a pin, or whatever other relevant nastiness the Attacker wants to do to the poor Defender. Attacker still has a standard action to make another CMB check to grapple and deal more damage. This second check is not to maintain the grapple, as that has already happened this round, and does not trigger any "on a successful check to maintain" effects.
Alternately: Attacker: Attempts, as a move action, to maintain the grapple but fails. They then use their standard action to attempt to maintain the grapple.
The crux of the problem is that you're not maintaining a grapple until your foe starts the turn grappled. From the grapple rules:
Once you are grappling an opponent, a successful check allows you to continue grappling the foe, and also allows you to perform one of the following actions (as part of the standard action spent to maintain the grapple).
#3: A creature with the Greater Grapple feat and the rake ability.
This is a combination of the first two examples, with a little added complexity. The Rake ability says, in part:
In addition to the options available to all grapplers, a monster with the rake ability gains two free claw attacks that it can use only against a grappled foe. ... 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.
First round the creature successfully grapples its target, generally as a result of an attack with the grab ability. It cannot rake this round, nor can it make additional grapple checks for the reasons outlined above.
At the start of the second round, the creature decides to attempt to maintain the grapple and succeeds on its check(s) and does terrible things to its victim. In addition, it gains its rake attack(s), albeit at a -2 due to the grappled condition, as a free action.
NB: You cannot make rake attacks without maintaining the grapple as rake requires your victim to be grappled and the grapple rules state "If you do not release the grapple, you must continue to make a check each round...". You could maintain, rake, and then drop the grapple, however.
#4: A creature with an attack with the grab ability, and an ability that triggers on successful grapple, such as constrict.
Abilities that trigger on a successful grapple check, such as constrict, don't change how the grapple flow works. Instead, they do things (generally damage) when a grapple check succeeds. This does include the initial check made when an attack with the grab ability hits.
As an extreme example, consider the humble giant octopus:
Melee bite +13 (1d8+5 plus poison), 8 tentacles +11 (1d4+2 plus grab)
Space 10 ft.; Reach 10 ft. (20 ft. with tentacle)
Special Attack constrict (tentacle, 1d4+2)
A typical full attack routine might look like this:
Tentacle attack against an opponent 20 feet away. The attack hits, allowing a CMB check to establish a grapple. This check succeeds and the food, er, victim take 2d4+4 damage (attack + constrict), and is moved to a space adjacent to the octopus. It can then take its remaining bite attack and 7 tentacle attacks at a -2 without constrict or decide to drop the grapple after each tentacle attack in order to potentially deal extra damage via constrict.
NB: The GM may limit the number of free actions you can take, especially with regard to repeatedly dropping grapples. You're best off consulting with them beforehand.
If it begins its turn still grappling its victim, it has a choice to make: either drop the grapple in order to repeat the above attacks or use a move action to make a CMB check to maintain the grapple. Without some ability like swallow whole, the octopus would deal far more damage doing the former than the latter. In the case of a creature with only one natural attack, such as a constrictor snake, maintaining the grapple, especially if it had the Greater Grapple feat, would be far more beneficial.