For the first question, which concerns just the smooth category without reference to metrics, the answer depends on the dimension of $M$. Before explaining this a small clarification is needed. By "a small neighborhood of $p$ and $q$" you probably mean a ball containing $p$ and $q$, otherwise one could take the neighborhood to consist of disjoint balls about $p$ and $q$, and then there would be no chance of finding a diffeomorphism $F$ with $F(p)=q$ and with $F$ the identity outside the neighborhood.
Let $Diff(D^n)$ be the group of diffeomorphisms $F:D^n\to D^n$ which are the identity in a small neighborhood $N$ of the boundary of $D^n$, with the $C^\infty$ topology on $Diff(D^n)$. The set $\pi_0Diff(D^n)$ of path components of $Diff(D^n)$ is the same as the set of isotopy classes of diffeomorphisms in $Diff(D^n)$ with the understanding that isotopies $F_s$ restrict to the identity on $N$ for all $s$. You are interested in the subspace $Diff(D^n;p,q)$ consisting of diffeomorphisms taking $p$ to $q$. It is not hard to see that every $F\in Diff(D^n)$ can be isotoped to take $p$ to $q$, and also that any isotopy $F_s$ between diffeomorphisms $F_0,F_1$ taking $p$ to $q$ can be deformed, fixing $F_0$ and $F_1$, to an isotopy with $F_s(p)=q$ for all $s$. In other words the natural map $\pi_0Diff(D^n;p,q)\to \pi_0Diff(D^n)$ is a bijection. (In fact this holds for all higher homotopy groups as well.)
If $\pi_0Diff(D^n)=0$, i.e., there is a single isotopy class, then the the answer to the first question is Yes. It is known that $\pi_0Diff(D^n)=0$ for $n=1$ (elementary), $n=2$ (Smale), and $n=3$ (Cerf). For $n=4$ it is still unknown whether $\pi_0Diff(D^n)$ is $0$ or not. For $n\geq 5$ it is known that $\pi_0Diff(D^n)$ is isomorphic to the group of exotic spheres of dimension $n+1$ (via the h-cobordism theorem and Cerf's pseudoisotopy theorem). This group is known to be $0$ for $n=5,11,55,60$ and it is known to be nonzero for all other $n\leq 125$ and all other even $n$, using hard calculations in the stable homotopy groups of spheres.
An embedding $D^n \subset M$ induces maps $\pi_0Diff(D^n)\to\pi_0Diff(M)$ and $\pi_0Diff(D^n;p,q)\to\pi_0Diff(M;p,q)$ by extending diffeomorphsms via the identity outside $D^n$. In these terms the question becomes whether the image of the map $\pi_0Diff(D^n;p,q)\to\pi_0Diff(M;p,q)$ is $0$. If $M$ is simply connected then it is not hard to show that the map $\pi_0Diff(M;p,q)\to \pi_0Diff(M)$ is a bijection, so in these cases the question boils down to whether the image of the map $\pi_0Diff(D^n)\to\pi_0Diff(M)$ is $0$. The answer is certainly Yes if $\pi_0Diff(D^n)=0$, so the answer is Yes for $M$ simply connected of dimension $n=1,2,3,5,11,55,60$. In the special case that $M$ is the sphere $S^n$ there is a classical elementary argument showing that the map $\pi_0Diff(D^n)\to\pi_0Diff(S^n)$ is injective (also on all higher homotopy groups), so this provides a No answer for $M=S^n$ for all other $n\leq 125$ and all even $n\geq 126$. For other manifolds $M$ the question of injectivity of $\pi_0Diff(D^n)\to\pi_0Diff(M)$ was studied in the heyday of differential topology but I can't recall any specific results or references at the moment.