1. If we do not assume that the diffeomorphism is identity outside a small ball, the answer is no. If $F$ preserves orientation but $F'$ reverses orientation, then there is no isotopy between $F$ and $F'$, but even if both diffeomorphisms preserve orientation there are counterexamples.
Here is one. Take $M=\mathbb{S}^1\times\mathbb{S}^1$ and $F:M\to M$ to be the identity. Let $F':M\to M$, $F'(z_1,z_2)=(z_2,\overline{z}_1)$ be the diffeomorphism that "changes the circles" (we take the complex conjugate - reflection in the $y$-axis to make sure that $F'$ preserves orientation). Fix $p=q=(1,1)\in\mathbb{S}^1\times\mathbb{S}^1$. Then $F(p)=F'(p)=q$. But there is no isotopy that would deform $F'$ to $F$, because $F$ and $F'$ are not homotopic. Indeed, $F(\mathbb{S}^1\times \{1\})=\mathbb{S}^1\times \{1\}$ but $F(\mathbb{S}^1\times \{1\})=\{1\}\times\mathbb{S}^1$ give two different elements in $\pi_1(M)$.
2. The answer is yes if $F$ is identity outside a small ball, but the isotopy is not smooth at the endpoint. We can assume that $F:\mathbb{B}^n\to\mathbb{B}^n$ is identity near the boundary, $p=0$ and $q\in \mathbb{B}^n$. According to our assumptions $F':\mathbb{B}^n\to\mathbb{B}^n$ be a diffeomorphism which is identity near the boundary and $F'(0)=q$. Then $(F')^{-1}\circ F$ maps $0$ to $0$. And $$ [0,1]\ni s \to F'(s((F')^{-1}\circ F)(\frac{x}{s})) $$ gives an isotopy between $F$ when $s=1$ and $F'$ when $s=0$. Moreover $p=0$ is mapped to $q$ for all $s\in [0,1]$. This isotopy is a continuous path of diffeomorphisms, but it is not smooth at $s=0$.
3. There are orientation preserving diffeomorphisms of $\mathbb{S}^6$ that are not isotopic to identity map. My understanding is that here we mean a smooth isotopy so the construction from the part 2 does not apply. Indeed, exotic $7$-spheres are obtained by gluing together two balls $\mathbb{B}^7$ along a diffeomorphism $F:\mathbb{S}^6\to\mathbb{S}^6$, see https://en.wikipedia.org/wiki/Exotic_sphere. If $F$ is smoothly isotopic to identity, then we obtain a standard sphere. Note that every orientation preserving diffeomorphism of $\mathbb{S}^6$ is isotopic to a diffeomorphism that is identity on the half sphere because near any point the difeeomorphism is close to a tangent map so this example applies to your question.
Remark. Regarding the construction in part 3, it is not my area of expertise to any comments are welcome.