zeroes of forms on Riemann surfaces Let $P$ be a point on a Riemann surface.
Does there exist a non-trivial differential form $\omega$ on $X$ such that $\omega$ vanishes at $P$?
Does there exist a non-constant rational function $f$ on $X$ such that the (rational) differential form $df$ vanishes at $P$?
What if we replace the set $\{P\}$ by a finite set of points?
 A: About  your first question:  
For  $g=\text {genus}(X)\geq 3$,  yes there exist non-zero differential forms  vanishing at an arbitrarily given point $P\in X$ : this results from Riemann-Roch applied to $\Omega_X(-P)$.
Indeed, writing $h^i=dim_\mathbb C H^i$, we have
$$h^0(\Omega_X(-P))=h^1(\Omega_X(-P))+1-g+\text {deg} \:\Omega_X(-P)=h^1(\Omega_X(-P))+1-g+2g-3   
\\\geq 0+1-g+2g-3=g-2$$   so that  $h ^0(X, \Omega_X(-P))\gt 0$ for $g\geq 3$, and thus there exists a nonzero differential form $\omega\in H^0(X, \Omega_X(-P))$ vanishing at $P$.
For $g=0$  no such form exists because only the zero form exists on $X$.
For $g=1$  a non zero differential form does not vanish, so your  question has a negative answer .
For $g=2$, I don't know what happens  .
A: Just to complete Georges's answer: let $S$ be an effective divisor on $X$ (i.e. a finite sum of closed points), the Riemann-Roch formula (with Serre duality) is
$$ h^0(Ω_X(−S))=h^0(O_X(S))+g-1-deg S\ge 1+g-1-\deg S=g-\deg S.$$
So the answer to your third question is positive when $\deg S\le g-1$ (this includes the case $S=P$ and $g=2$). 
Things become more interesting when $\deg S=g$. The above arguments show that $h^0(\Omega_X(-S))> 0$ if and only if $h^0(O_X(S))\ge 2$. One can identifiy the set of effective divisors of degree $g$ on $X$ to the symetric product $X^{(g)}$ (which is $X^g$ quotient by the symetric group in $g$ elements acting on $X^g$ by permutation of the coordinates). Fix a point $x_0\in X$, then we have a morphism to the Jacobian of $X$
$$ f: X^{(g)} \to J(X), \quad S \mapsto [S-gx_0].$$ 
It is well known that $f$ is birational. But what is the exceptional locus of $f$ ? It consists in those $S$ such that $\dim f^{-1}(f(S))>0$. As $f^{-1}(f(S))=|S|$ the linear system of effective divisors linearly equivalent to $S$ and has dimension (as projective variety) $h^0(O_X(S))-1$,  we see that 

when $\deg S=g$, $h^0(\Omega_X(-S))> 0$ if and only if $S$ belongs to the exceptional locus of $f$.

Edit Answer to Question 2. Let $t$ be a rational function which is an uniformizing element of $X$ at $P$, let $f=t^2$. Then $df=2tdt$ has a simple zero at $P$. 
Remark on the divisor of a rational section (to respond to a question raised in the comments). Let $L$ be an invertible sheaf on $X$, let $s$ be a non-zero rational section of $L$ (i.e. $s\in L(U)$ for some non-empty open subset $U$ of $X$ and $s\ne 0$). Then the divisor $\mathrm{div}_L(s)$ is a Cartier divisor such that $O_X(\mathrm{div}_L(s))\simeq L$. 
This is can be check by writing $L$ on an open covering $\{ X_i\}_i$ of $X$ as $L|_{X_i}=e_i O_{X_i}$. So $s=e_if_i$ for some $f_i\in K(X)$. The Cartier divisor $\mathrm{div}_L(s)$ is represented by $\{ (X_i, 1/f_i)\}_i$ and we have $sO_X(\mathrm{div}_L(s))=L$.
In the particular case when $L=\Omega_X$, for any rational differential form $s$, we have $\mathrm{div}_L(s)\simeq K_X$ for any canonical divisor $K_X$ on $X$. 
