In order of your questions: Not exactly, No, and No.
1. You don't lose feats, although some of them may become unusable.
The bonus feat you received for being a human is not lost. Under the description of feats:
A character can't use a feat if he loses a prerequisite, but he does not lose the feat itself. If, at a later time, he regains the lost prerequisite, he immediately regains full use of the feat that prerequisite enables.
In Pathfinder, the polymorph effect causes a creature to take on the physical form of another creature, but it does not fully "become" that creature. A human polymorphed into a squirrel is still technically a human. I strongly suggest reading through the full description of what is changed by a Polymorph effect.
2. Polymorphing does not inherently change your race, and so racial bonuses are not necessarily lost.
While under the effects of a polymorph spell, you lose all extraordinary and supernatural abilities that depend on your original form (such as keen senses, scent, and darkvision), as well as any natural attacks and movement types possessed by your original form. You also lose any class features that depend upon form, but those that allow you to add features (such as sorcerers that can grow claws) still function.
The human's Bonus Feat racial trait, Skilled racial trait, and +2 racial bonus all remain, because none of them meet the criteria above. While polymorphed, you are still inherently a human, even though you don't look like one. You will temporarily lose access to your 30 foot movement speed, and replace it with the movement speed(s) of your new form.
Your ability scores are not modified by this change unless noted by the spell.
Your character's Strength, Dexterity, and Constitution may be temporarily adjusted with Size bonuses. The rules page that I linked has a table with these adjustments, although these bonuses may depend by the source of the Polymorph effect. Usually, growing tends to cause Size bonuses to Strength and Size penalties to Dexterity, and shrinking tends to do the opposite.
For comparison, take a look at the Reincarnate spell; it actually does change a creature's race, causing them to change their racial ability score modifiers.
3. Polymorphing does not cause you to gain any feats.
The creature's bestiary page refers to a typical version of that creature; you won't acquire the squirrel's feats just by polymorphing into one.
A polymorph spell transforms your physical body to take on the shape of another creature. While these spells make you appear to be the creature, granting you a +10 bonus on Disguise skill checks, they do not grant you all of the abilities and powers of the creature.
The squirrel's racial bonus feats exist because it is inherently a squirrel. If the squirrel was polymorphed into a human form, it would have its original feats, ability scores, and skills. Due to the size increase, its Strength would get a Size bonus and its Dexterity would get a Size penalty.
They can take any race-restricted character option
As pointed out in this answer, in Pathfinder, most half-breeds of humans are also treated members of both parent races, and thus, also treated as humans. This rule extends to:
Orc Blood: Half-orcs count as both humans and orcs for any effect related to race.
Elf Blood: Half-elves count as both elves and humans for any effect related to race.
- Aasimars with Scion of Humanity racial trait:
Scion of Humanity: Some aasimars’ heavenly ancestry is extremely distant. An aasimar with this racial trait counts as an outsider (native) and a humanoid (human) for any effect related to race, including feat prerequisites and spells that affect humanoids.
Such a tiefling doesn’t need to succeed at a Disguise check to appear to be human and count as humanoid (human) as well as outsider (native) for all purposes (such as humanoid-affecting spells like charm person or enlarge person).
Such geniekin appear to be human, save perhaps minor features like unusual eye color, and they count as humanoid (human) as well as outsider (native) for all purposes (such as humanoid-affecting spells such as charm person or enlarge person).
Fetchlings with this racial trait count as outsiders with the native subtype and humanoids with the human subtype for any effect related to race, including feat prerequisites and spells that affect humanoids.
You count as both human and your race for the purpose of taking character options, such as archetypes, feats, spells, traits, and prestige classes.
Keep in mind that planar half-breeds exist for non-human races, as pointed out in Blood of Angels (Aasimars) and Blood of Fiends (Tiefling). Such variants would be an exception to this, as pointed out on the sidebar of Blood of Angels, for instance:
Non-human aasimars have the same statistics as human aasimars with the exception of size. Thus a halfling aasimar is Small but otherwise possesses the same statistics and abilities as a human aasimar-the difference is purely cosmetic. Non-human aasimars do not possess any of the racial abilities of their base race. However, they are usually raised in the same cultural context as other members of their base race, and thus generally adopt the same fighting style as their peers, use the same types of weapons and armor, and study the same skills.
So from a rules-as-written perspective, an Aasimar that has an elf parent and takes Scion of Humanity would count as humanoid(native) and still count also as humanoid(human) instead of humanoid(elf).
But are also affected by any race-specific effects
This may sound strong, but keep in mind that this also affects race-specific effects or effects based on creature type, like the Bane special weapon property, or an Intelligent Magic Item that wishes to defeat/slay all members of a specific race.
A half-orc, for example, is both a humanoid(orc) and a humanoid(human) creature. Meaning that they are affected by both a Human-Bane weapon and a Orc-Bane weapon (luckily, bane doesn't stack). A half-elf would meet the requirements of crafting a Cloak of Elvenkind. And so on.
Best Answer
There is no rule that allows you to take a Drawback to gain a Feat.
It's not a nerf if the rules don't support it and you don't allow it.
Drawbacks are very clear:
A couple things to note include
Furthermore, Traits are valued at two traits being roughly equivalent to one Feat. This isn't exactly true, but that's the valuation based on the Additional Traits feat. You'll also note, that you cannot take the Feat more than once. By this logic, if you allowed it (which is not supported by any rules), a player could take 2 Drawbacks for 1 Feat.
Side Note: Major Drawbacks are an option from 3rd party producer Rogue Genius Games. You are under no obligation to include these rules in your game. It makes no mention of doing this more than once on a character.
GM Fiat isn't a bad thing.
Don't let your problem player try to throw this in your face. The Gamemaster's primary responsibility is
If you're Gamemastering, you get to pick which rules and non-rules apply to your game. No player gets to pick your rules. If they don't like it, then there needs to be an open discussion where you explain that 'no, you cannot give yourself bonus feats; I don't want to have to manage that many Drawbacks to actually make you pay for them, and there's no rule that says you could anyway'. If they're worried that you're trying to manage their power level... tell them yes. Yes you are, because it's your job and 1st/2nd level characters aren't supposed to have 5 Feats.