If an $\mathbb{F}_p$-point is smooth, then it has Hensel lift In  the writeup(http://www.crm.umontreal.ca/sms/2014/pdf/stoll.pdf), the following statement has been made on page 2, under the proof of Proposition 3 :

If $p + 1 > 2g
\sqrt p$ and $p$ does not divide the discriminant of $f$, then $C$
  reduces to a smooth hyperelliptic curve of genus $g$ over $\mathbb{F}_p$, which by the Weil bounds has $\mathbb{F}_p$-rational points (and all these points are smooth). By Hensel’s Lemma, a smooth $\mathbb{F}_p$-point lifts to a point over $\mathbb{Q}_p$.

As the introduction section says:

A hyperelliptic curve $C$ over a field $k$ not of characteristic 2 is the smooth projective curve associated to an affine plane curve given by an equation of the form 
  $$y^2 = f(x)$$,
  where $f$ is a square-free polynomial of degree at least 5. If the degree of $f$ is $2g + 1$
  or $2g + 2$, then the curve has genus $g$. 

I am looking for an answer/hints towards the following questions


*

*How does C reduce to the smooth hyperelliptic curve over $\mathbb{F}_p$?

*Which version of Hensel's Lemma is used here? Specifically, how does "smooth" play a role in this case?

*Finally, I'm hoping someone could point me to a proof of this version of Hensel's lemma so as to understand the "$p + 1 > 2g
\sqrt p$ and $p$ does not divide the discriminant of $f$" part.

 A: Others can (and I hope will) give better answers than mine, but until they jump in, I can get you started. This answer will certainly be a hack-job.
I’m assuming that the curve is originally given as a variety (scheme, etc.) over $\Bbb Z_p$, the ring of $p$-adic integers. As far as I’m concerned, all bets are off if there are nonintegral coefficients. Now, the discriminant is a polynomial in the coefficients of $f(x)$, all of these in $\Bbb Z_p$, and the same formula for discriminant applies in characteristic $p$. So if $p$ does not divide the discriminant as a $\Bbb Z_p$-element, then the discriminant formulas give you a nonzero element of $\Bbb F_p$, telling you that your reduced curve (over $\Bbb F_p$) is nonsingular. Degree tells you the genus.
About lifting and how Hensel lets you lift a point that’s nonsingular, I have a good hands-on feel for the situation, since I’ve done it many times. But you’ll have to wait for the knowledgeable ones’ answer for a real explanation. I think, though, that if you look at a curve like $y^2=x^3+px+1$ and try to lift the $\Bbb F_p$-point $(2,3)$, you’ll see how nonsingularity comes in: there certainly is a point $(2+\varepsilon,3+\delta)$ of this elliptic curve over $\Bbb Z_p$, where $\varepsilon,\delta\in p\Bbb Z_p$. To see a case of a nonsmooth point, look at the zero-dimensional case of $X^2-p=0$. In characteristsic $p$, it becomes $X^2=0$, a nonsmooth point; a lifting would be a $\Bbb Z_p$-square root of $p$, which existeth not.
About your last uncertainty, the condition $p+1>2g\sqrt p$, I’m not at all sure; maybe it has something to do with guaranteeing that there are $\Bbb F_p$-rational points on your curve.
