The physical form will be your limiting condition
The Players Handbook, Page 67 states (emphasis mine):
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can’t use any o f your special senses, such as darkvision, unless your new form also has that sense.
Racial Traits that are tied to physical form requirements would not transfer to WildShape. An example of this is the Aarakocra Trait of Flight(because wings). If you Wildshape into a beast that can't fly or doesn't have wings, then your Racial Trait wouldn't transfer over.
A trait like Elf Fey Ancestry for advantage on savings throws against being Charmed and immunity to magical sleep would be something that would cross over to your Wild Shape as it has no specific physical origin.
If you are unsure or want to verify, talk about it with your DM or player. The 'physical' guideline has enough gray areas that the less obvious traits may need a table ruling.
Crawford agrees in an unofficial tweet:
A racial trait works with Wild Shape unless that trait requires anatomy the beast form lacks.
Yes, they would still benefit from Expertise in Wild Shape
As stated in the part of the description of the druid's Wild Shape feature you quote:
You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature’s bonus instead of yours.
In addition, another bullet point in the Wild Shape feature description says:
You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so. However, you can’t use any of your special senses, such as darkvision, unless your new form also has that sense.
So, to see whether your Expertise class feature is retained, we need to look at the description of the rogue's Expertise feature:
At 1st level, choose two of your skill proficiencies, or one of your skill proficiencies and your proficiency with thieves’ tools. Your proficiency bonus is doubled for any ability check you make that uses either of the chosen proficiencies.
Nothing in the feature description specifies a particular anatomy needed to benefit from Expertise (...it'd be weird if it did, honestly). Thus, you do retain the benefit of the Expertise feature.
Putting it all together: You still benefit from the Expertise feature in Wild Shape. You retain all your skill proficiencies in Wild Shape, in addition to those of your new form. And if the creature is also proficient in the same skill, you use whichever total bonus to the skill is higher between yours and the beast's (for a skill you have Expertise in, your own bonus will likely be higher).
Note: Your own modifier for a skill may change depending on the ability score that the skill is associated with. Per the Wild Shape description:
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores.
Thus, your own Strength, Dexterity, and Constitution scores are replaced by those of your new form while in Wild Shape. For those skills that are tied to these three scores (or rather two, since Constitution isn't associated with any skills), your modifiers are also changed accordingly.
For instance, if your own Strength modifier is +1 but your new form's Strength modifier is +5, your Athletics skill modifier will increase by 4 - assuming the beast is not also proficient in Athletics with a higher modifier (in which case you would use the beast's Athletics modifier instead of yours).
Correspondingly, your modifier for a skill may actually decrease if the new form is not proficient in the skill and its corresponding ability score is lower. For instance, if you have a Dexterity modifier of +3, and the beast's Dexterity modifier is -2, your modifier for the Stealth skill would actually decrease by 5.
You can determine your new modifiers for the skills associated with Strength and Dexterity by recalculating them based on your new ability score, and then comparing them to the beast's statblock. If the beast is not proficient, you use your own recalculated modifier. If it is proficient, then you compare your recalculated modifier to the modifier given in the statblock; use whichever of the two is higher.
Best Answer
Yes, if you have fangs.
You've already rightly pointed out that you have "appropriate fangs", but I wanted to expound of this a little bit to make sure future askers of this similar question understand that caveat. You need fangs to make the bonus action unarmed strike.
And then
So, yes, you CAN make an unarmed strike in wild-shape, as pointed out in this answer. But here's the catch: you don't have elongated fangs unless you assume a form that has elongated fangs. Wildshape limits your racial features if your new physical form is not capable of doing so. Furthermore, you adopt a new physical form, so you wouldn't have your previously elongated fangs, you would have new teeth. If you transform into a horse, you don't have elongated fangs.
So what this means is that you can't use your bonus action to make an unarmed strike because you don't have the prerequisite (using your elongated fangs). And even if you could, you wouldn't get the elongated fang damage because you also didn't hit "With your fangs".
But lets say you DO have fangs, like the aforementioned tiger. Then you absolutely could make that unarmed strike with fang damage, provided you were Shifted before you made the attack.
What are "Elongated Fangs"?
The "Elongated" fangs are relative to your humanoid form.
So you have sharp canines as a humanoid, but they can't really bite things that well. When you shift, they are elongated. So as long as your new form has fangs, like most predators do, then you can use them. Don't think of elongated fangs as these huge dagger-like, snake-like fangs, but rather just elogated in relation to your humanoid canines (which are already pronounced). A tiger, for example, would have fangs that are elongated in relation to your humanoid fangs. Your DM will decide if your fangs are good enough to use, since the rules don't tell us which animals have fangs nor how long they need to be. We have to use our best judgement.