Edit: there was a mistake in my calculation. I hope I fixed it now. More edits: simplified the computation.
The curve $C_{c,p}$ over $\mathbb{F}_p$ defined by $x^2+xy+y^2=c$ is a conic, so if it is smooth and absolutely irreducible its completion in the projective plane will contain $p+1$ points. But we can also go another route: over the field $\mathbb{F}_{p^2}$, we can factor the equation as $(x-y\omega)(x-y\omega^2)=c$, with $\omega$ a primitive third root of unity (which is not contained in $\mathbb{F}_p$ since $p \equiv 2 \pmod{3}$). Now if $x,y \in \mathbb{F}_p$, notice that the product $(x-y\omega)(x-y\omega^2)$ can be written $(x - y\omega) \cdot \operatorname{Frob}(x-y\omega)$, in other words this is the field norm of the element $x-y\omega$ for the field extension $\mathbb{F}_p \subset \mathbb{F}_{p^2}$, so we are in fact looking for the number of elements $x-y\omega \in \mathbb{F}_{p^2}$ such that their norm equals $c$. The norm of an element $\xi$ equals $\xi \cdot \operatorname{Frob}(\xi)=\xi^{p+1}$, so we are really solving the equation $\xi^{p+1}=c$ over $\mathbb{F}_{p^2}$. It is easy to see this has $p+1$ solutions: the multiplicative group $\mathbb{F}_{p^2}^{\times}$ is cyclic of order $(p+1)(p-1)$, whereas the order of $c$ must divide $p-1$ because $c \in \mathbb{F}_p^{\times}$. Therefore the answer to the first question is that there are $p+1$ solutions.
The second question has for its answer $(p+1-a)/2$, where $a$ is the number of solutions of $x^2+xy+y^2=c$ with $x=y$. Evidently we have $a=\chi(3c)+1$, so that the answer to the question is $(p-\chi(3c))/2$.
For the third question, first note the product is non-zero since $x^2+xy+y^2=0$ would imply $(x-y\omega)(x-y\omega^2)=0$, contradicting the existence of a primitive third root of unity in $\mathbb{F}_p$. Then, since $f(c)$ equals $(p-\chi(3c))/2$, the exponent in the formula $\prod_c c^{f(c)}$ which you found really assumes only two values, namely $(p+1)/2$ and $(p-1)/2$. So if we denote by $A$ the set of $c$ such that $\chi(3c)=-1$, and by $B$ the set of $c$ such that $\chi(3c)=1$, then the product $\prod_c c^{f(c)}$ equals $\prod_{c \in A} c^{(p+1)/2} \prod_{c \in B} c^{(p-1)/2}$. Note that $\chi(3) = (-1)^{(p-1)/2} \left( \frac{p}{3} \right) = -(-1)^{(p-1)/2}$ since $p \equiv 2\pmod{3}$. So, by the multiplicativity of $\chi$, we can also describe $A$ as the set of values $c$ such that $\chi(c)=-\chi(3)=(-1)^{(p-1)/2}$, so that $B$ equals the set of values $c$ with $\chi(c)=(-1)^{(p+1)/2}$. We will evaluate the two factors of the product as follows:
The easiest thing to do is rewrite the expressions over which the products are taken in terms of a generator $g$ of the multiplicative group of $\mathbb{F}_p$. An element $c \in A$ has the general form $g^{(p-1)/2} g^{2k} = -g^{2k}$, since we said that $A$ is the set of values $c$ such that $\chi(c)=(-1)^{(p-1)/2}$, and where we have used $\chi(g)=g^{(p-1)/2}=-1$. So the product indexed by $A$ then becomes $\prod_{k=1}^{(p-1)/2} (-1)^{(p+1)/2} g^{k(p+1)}$. Likewise, the general form of an element of $B$ is $g^{(p+1)/2} g^{2k} = -g^{2k+1}$, making the product indexed by $B$ equal to $\prod_{k=1}^{(p-1)/2} (-1)^{(p-1)/2} g^{(2k+1)(p-1)/2}$.
So taking the two products together, we are reduced to the computation of the product $\prod_{i=1}^{(p-1)/2} -g^{k(p+1)} g^{(2k+1)(p-1)/2}$, which equals $\prod_{i=1}^{(p-1)/2} -g^{k(p+1)} g^{k(p-1)} g^{(p-1)/2}$. The minus sign cancels the factor $g^{(p-1)/2}=-1$, so this also equals $\prod_{i=1}^{(p-1)/2} g^{2kp}$, which equals $g^{\sum_{i=1}^{(p-1)/2} 2kp} = g^{p(p^2-1)/4}.$ Using $g^p=g$ and once again that $g^{(p-1)/2}=-1$, we finally get that this equals $(-1)^{(p+1)/2}$, which equals $-\left( \frac{-1}{p} \right)$.
