No.
The key here is that the Monk weapon ability doesn't actually add the keyword "finesse" to the weapon, it just lets you use Dex for attack and damage.
Defensive Duelist specifies that a weapon have that keyword, and thus, no, it can't be used.
Adding more fuel to this fire is Jeremy Crawford talking about natural weapons and coming down on the side that just because you are attacking with Dex does not mean you are adding finesse:
No. A weapon has the finesse property only if its description says so, and using Dex. to hit doesn't equal finesse.
Would this be a game breaking houserule? No, that's quite unlikely. The Staff does d6 or d8 damage, and there is already a finesse weapon that does d8 (Rapier). The monk isn't proficient in that, so it's not available, but it's not out of bounds for them to be able to use it.
Note: Brian points out that two monk weapons are finesse weapons, and as such would be usable with this combination (and get the Monk's proper monk weapon die). These are the dagger and the short sword.
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.
Best Answer
First, the specific case of Alter Self. The Natural Weapons feature of Alter Self explicitly modifies your Unarmed Strike. So if your Unarmed Strike is not a finesse weapon, that doesn't change. This is covered pretty thoroughly in this answer.
However, the general case of 'Do natural weapons have the Finesse property?' is much less clear cut. Take the Raven (Appendix A of the Monster Manual). The Raven has a Str of 2 (modifier -4), a Dex of 14 (modifier +2), and, from the 'Proficiency Bonus By Challenge Rating' table (page 8 of the Monster Manual), its proficiency bonus is +2. Its attack bonus with it's natural weapon (beak) is +4. So it seems clear that it is using
There are numerous examples along the same lines, but I chose the raven because the difference between its Str and Dex makes it pretty clear that it's using Dex.
On the other hand, we have the Mastiff. (Also Appendix A of the Monster Manual). It has a Str of 13 (modifier +1), a Dex of 14 (modifier +2), and the same +2 proficiency bonus. Its attack bonus is +3. So it appears to be using
The Mastiff is not nearly as good an example, but every creature I could find with much higher Dex than Str uses Dex to attack.
Based on these (and a lot of other) examples, it seems clear that some creatures can use Dex to attack with their natural weapons, but some cannot. Alternatively, it's possible that creature attack bonuses are either arbitrary or not calculated the same way as humanoid attack bonuses are, but if this is the case there is no way to answer this question outside of the specific case of Alter Self.