So, for the first one I don't know what I have to do to prove that it is a map, it seems clear that it is, as if $f$ is defined for every set in $P_1$ to every set in $P_2$, the inverse would be a map as well, moreover, we can take "some" (the down-set) of those elements in $P_2$, the inverse function will be defined for it. I would appreciate the input on wheather my reasoning is correct, and if it is id like some input on how to translate that into formal proof. If it is not, I would like to know how to do it.
I don't understand what you're saying so I can't comment on the correctness of your reasoning. I will however comment on the part where you say it is clear that $g$ is a map.
Regardless of how clear that is, you're being asked to prove more than $g$ being a map, you're being asked to prove that $g$ is a map from $\mathcal O(P_2)$ to $\mathcal O(P_1)$, so you need to prove that for every $Z\in \mathcal O(P_2)$, it holds that $f^{-1}[Z]$ is a down-set of $P_1$.
To prove this take $x,y\in P_1$ such that $y\leq x$ and $x\in f^{-1}[Z]$. The goal is to prove that $y\in f^{-1}[Z]$. By definition $y\in f^{-1}[Z]\iff f(y)\in f[Z]$. So it suffices to prove that $f(y)\in Z$. To prove this you'll need the fact that $Z$ is a down-set and $f$ is order preserving.
Now you've earned the right to write $g\colon \mathcal O(P_2)\to \mathcal O(P_1)$.
For the second part, call $O$ the set of down-sets. First, assume $f$ is order embedding, so for all $x,y \in P_1 : \ x \le y \iff f(x) \le f(y)$.
Let $Z \in O(P_1)$, because it is a member
Assume $g$ is onto, then the range is $O(P_1)$. The inverse of an order preserving map is order preserving.
It's an equivalence, you seem to be starting by $\implies$, so you need to prove that $g\colon \mathcal O(P_2)\to \mathcal O(P_1)$ is onto. In this respect you just typed "Let $Z \in O(P_1)$, because it is a member" and then you started proving the direction $\Longleftarrow$, as is suggest by the sentence "Assume $g$ is onto". Here too you don't seem to have proved what's required, namely that $f$ is an embedding. And you talk about the inverse without knowing if $f$ is injective, not every order-preserving map is injective.
To prove $\implies$, assume $f$ is an embedding. You wish to prove that $g$ is surjective, that is that $\forall Y\in \mathcal O(P_1)\exists Z\in \mathcal O(P_2)\left(g(Z)=Y\right)$. Let $Y\in \mathcal O(P_1)$. It's natural to try $Z':=f[Y]$. Is $Z'$ a down-set? Not necessarily. Try $Z:=\mathord{\downarrow} Z'$ instead. Does it hold that $g(Z)=Y$? In other symbols, does it hold that $f^{-1}\left[\mathord{\downarrow} f\left[Y \right]\right]=Y?$ Try to prove it. You'll need to use the fact that $f$ is an embedding.
To prove $\Longleftarrow$, start by supposing that $g$ is onto. It's already given that $f$ is order-preserving so all that is left is to prove that $\forall x,y\in P_1(f(x)\leq f(y)\implies x\leq y)$. Let $x,y\in P_1$ be such that $f(x)\leq f(y)$. Consider the down-sets $\mathord{\downarrow} x$ and $\mathord{\downarrow} y$. Since $g$ is onto, there exist down-sets $Z_x, Z_y\in \mathcal O(P_2)$ such that $g\left(Z_x\right)=\mathord{\downarrow} x$ and $g(Z_y)=\mathord{\downarrow} y$. Note that $f(x)\in Z_x$, prove that $f(x)\in Z_y$, infer that $x\in \mathord{\downarrow} y$ and conclude that $x\leq y$.
For the third, Assume $f$ is onto $P_2$, then $ran(f)=P_2$. In search of a contradiction, assume $g$ is not one to one. My idea is: if $g$ is not one to one, then 2 elements of $O(P_2)$ can go to the same element in $O(P_1)$ so when i take the inverse, $f$... i have a problem. I was thinking about setting $a < c \in O(P_2)$ and $g(a) = g(c)$, but i don't know how else to procede.
Your try here seems legitimate. In fact, because $f$ is surjective, it is also injective so you're allowed to talk about the inverse of $f$. In any case I don't see how to make your proof work without doing and I do below (which avoids contradiction).
For $\Longrightarrow$, assume $f$ is onto. The goal is to prove that $g$ is injective, i.e., that for all $Z_1, Z_2\in \mathcal O(P_2)$ it holds that $f^{-1}\left[Z_1\right]=f^{-1}\left[Z_2\right]\implies Z_1=Z_2$. Since $f$ is surjective and $f^{-1}\left[Z_1\right]=f^{-1} \left[Z_2\right]\implies f\left[f^{-1}\left[Z_1\right]\right]=f\left[f^{-1}\left[Z_2\right]\right]$, this is actually an elementary set theory result which has nothing to do with order and it has been done thousands of times before, see it for instance here.
For $\Longleftarrow$, one possibility is to prove it by contraposition, that is, suppose that $f$ isn't surjective and prove that $g$ isn't one-to-one. Well, assuming $f$ isn't surjective, there exists $z\in P_2$ which isn't in the image of $f$. Now consider $\mathord{\downarrow} z$ and $\left(\mathord{\downarrow} z\right)\setminus \{z\}$ and show that $g\left(\mathord{\downarrow} z\right)=g\left(\left(\mathord{\downarrow} z\right)\setminus \{z\}\right)$ (you'll need the fact that $z$ isn't in the image of $f$). This finalizes the proof.
