The Setup

Abe, a human Clr4 of Nobanion, has been tasked by his superiors to apprehend a man-eating tiger. He expended his spells traveling to the tiger's lair. The tiger, smelling the meat juice Abe bathed in before his trip, thinks Abe a potentially tasty snack. (Note: The tiger is correct.)

Round 1

Abe is between 10 ft. and 80 ft. from the tiger, which has an unimpeded route to him.

1. Abe and the tiger make initiative checks. The tiger goes first.
2. The tiger takes a full-round action to make a charge.
3. Due to its special ability pounce, at the end of the charge the tiger makes a full attack.¹
4. The tiger makes an attack with its claw, hitting and dealing damage.
5. Due to its special ability improved grab, the tiger attempts to start a grapple without provoking an attack of opportunity and without needing to make a touch attack, hence advancing immediately to Step 3: Hold.
6. The tiger and Abe make opposed grapple checks.
7. The tiger wins, getting a hold on Abe. Due to the tiger's special ability improved grab, the tiger does not deal its unarmed strike damage, and the tiger moves Abe into its space.

8. Because If You're Grappling says

   If your base attack bonus allows you multiple attacks, you can attempt one of these actions in place of each of your attacks, but at successively lower base attack bonuses.

   the tiger's base attack bonus means it can't make more attacks this way and is unable to use immediately, for example, the options Attack Your Opponent or Damage Your Opponent.

9. The tiger makes two rake attacks against Abe. The tiger's special ability pounce says that

   If a tiger charges a foe, it can make a full attack, including two rake attacks.

   Normally, rake attacks have all kinds of restrictions placed on them, but when a tiger charges and makes a full attack, that full attack includes the two rake attacks, no matter the usual limitations of rake, specific overriding general. The tiger, despite having grappled its foe this turn, has charged and made a full attack, so it makes its rake attacks.²

   Note: This is where the argument starts. A player whose character gets raked in the first round by a tiger grappling him will undoubtedly, if allowed, flip to the Monster Manual on rake, read that, and argue in favor of not getting raked. It's up to the DM if general should instead override specific here, but doing so substantially weakens the tiger as a threat, and the tiger's pounce is awfully specific. The only comparison I can make is that if the tiger's pounce said the tiger could instead, for example, breathe fire or sting a creature after that including, would the player be okay with that? The specific rules would be overridden in those cases, too. But I can easily imagine a dispirited player whose character was just tiger-mauled taking exception to the printed text, claiming the DM made a bad call, and departing the table in a huff. In a tigercentric campaign, the DM should establish early whether the pouncing-raking tiger reading is the correct one for his table, and, if it is, the DM should make sure the players know ahead of time that, perhaps, tigers are not to be trifled with by low-level characters.

On his turn, Abe takes a standard action to attempt to escape the tiger's grapple. He and the tiger make opposed grapple checks. The tiger wins.³

Round 2

The tiger decides it wants to beef up its reputation as a man-killer and tries to kill Abe.

1. The tiger and Abe make opposed grapple checks, the tiger taking the option Damage Your Opponent.
2. The tiger wins. According to the special ability improved grab, as the tiger has made a successful grapple check, it deals its claw damage as that's the attack that established the hold. Because of its successful grapple check, it also deals Abe 1d4+6 nonlethal damage, equivalent to its unarmed strike damage. Note that the tiger still lacks a base attack bonus sufficient to make multiple attacks this way during a grapple. Further, the tiger could've gotten cocky and suffered a -4 penalty to the opposed grapple check when it took the Damage Your Opponent option to deal Abe lethal damage instead of nonlethal damage.
3. As this is the second round of grappling, the tiger can now make two rake attacks normally.

The tiger will likely continue this process until Abe or it is unconscious, dead, or something (possibly divine) intervenes.

Notes
¹ Most DMs rule that the full attack replaces instead of supplements the attack at the end of the pounce.
² This exception was slightly clearer in the Monster Manual for Dungeons and Dragons, Third Edition: "If the tiger pounces on an opponent, it can also rake," but this really only clarifies things a little bit.
³ Note that the tiger made a grapple check, and, according to the special ability improved grab, "each successful grapple check [the creature] makes during successive rounds automatically deals the damage indicated for the attack that established the hold" (emphasis mine). Luckily for Abe, it's still the same round in which the hold was established.

Replacing a claw attack with an unarmed strike

Sorry, but no. Unarmed strikes use a weird hybrid of the rules for manufactured and natural weapons, but for the purposes of full-attacks, they work like manufactured weapons. That is, you get iteratives with them, but if you can only combine them with natural weapons by making those natural weapons secondary (−5 attack penalty, only ½Str to damage).

The first rule that you quote is specifically about spells and effects. A full-attack is not either of those.

Claws and lack of offhand unarmed strikes

Feral Combat Training does mean that anything from the monk’s unarmed-strike-improving class features can apply to natural weapons, and that can include the bit about never being offhand.

However, claws and other natural weapons are never “offhand” to begin with. The term “offhand” only applies when using two-weapon fighting, and that combat option does not interact with natural weapons (aside from the attack penalty, which applies to all attacks). So the fact that the monk class feature, combined with Feral Combat Training, says that natural weapons are never offhand does not do anything because that was already true.

Instead of “main hand” and “offhand,” natural weapons are either “primary” or “secondary.” These are different. When combined with manufactured weapons (or unarmed strikes) in a given full-attack, all natural weapons are secondary: they receive the −5 penalty and get only ½Str to damage. Neither the monk class nor Feral Combat Training does anything about treating them as secondary or removing or reducing the penalties for being secondary.

So whether you have Feral Combat Training or not, your full-attack using unarmed strikes is:

Unarmed Strike, Claw (−5), Claw (−5), Claw (−5)

If you have Feral Combat Training, the claws do benefit from the improved base damage dice of unarmed strike, however, even if they’re still stuck with ½Str to damage.

Two-Weapon Fighting, Feral Combat Training

If you are actually using two-weapon fighting, the provision about monks never having offhand unarmed strikes meaningfully applies only to the unarmed strike. It “applies” to the claws, but does nothing for them.

So, for example, if your two weapons are a sai and an unarmed strike, and you have the Two-Weapon Fighting feat, your attack routine would be:

Sai (−2), unarmed strike (−2), claw (−5), claw (−5)

The unarmed strike would add your full Strength to its damage, however. Note that I assumed that the sai took up one of your claw-hands. I did not wish to get into the debate about whether one can use two unarmed strikes as part of two-weapon fighting.

You didn’t ask, but about Flurry of Blows

All of the statements above about full-attack apply equally well to flurry of blows, except that you need Feral Combat Training to use natural weapons in a flurry at all, and flurry of blows cannot be combined with two-weapon fighting because of Paizo nonsense.

Personal recommendation

For the record, monks, natural attacks, and how they combine, these are some of the worst things in Pathfinder. The rules are confusing, complicated, and the result works very poorly. I suggest you save yourself a headache and just... not.