Some extended question on bounds of Rational map

Question

I saw this question here: I am stuck on the same kind of question but my problem is a bit more general which thrives me to post a new one. I am copying a bit definition from that post to save some typing time.

Let $\phi: \Bbb P^1 \to \Bbb P^1$ be a rational map then we define $Aut(\phi)=\{f \in PGL_2(\Bbb C): \phi^f=f^{-1}\phi f(z)=\phi(z)\}$

Here, in general, the definition of a rational map is: Let $\mathbb{P}^n$ and $\mathbb{P}^m$ be projective spaces. If $m$ homogeneous polynomials in $n+1$ variables of the same degree "d" give a partially defined map from $\mathbb{P}^n$ to $\mathbb{P}^m$ then this map is called a rational map. We can exactly define this on a projective variety as well.

Let $\phi: \Bbb P^1 \to \Bbb P^1$ be a rational map of degree $d \geq 2$. Then $Aut(\phi)$ is a finite subgroup of $ PGL_2(\Bbb C)$ and its order is bounded by a function of $d$.

Proof: Let $f \in Aut(\phi)$ . Then for any point $P \in \Bbb P (\Bbb C)$ and any $n\geq 1$ we have $$\phi^n(P) = (\phi^f)^n(P ) = (f^{-1}\phi^nf) (P )$$, and hence $$f (\phi^n(P )) = \phi^n(f( P))$$.

In particular, if P is a periodic point of (primitive) period n, then f (P) is also a periodic point of (primitive) period n. Thus each $f \in Aut(\phi)$ induces a permutation of each of the sets $Per_n (\phi)$ and $Per_n^{**} (\phi )$, where $Per_n (\phi)$ and $Per_n^{**} (\phi )$ are the collection of periodic point and premitive peiodic points of period $n$. [so $P\in Per_n (\phi)$ means $\phi^n(P)=P$ and $P\in Per_n^{**} (\phi)$ means n is the smallest natural no such that $\phi^n(P)=P$ holds.

Choose three distinct integers $n_l, n_2, n_3$ such that $\phi$ contains primitive n periodic points for each value of n. Now we have a result that says that we can find such values, and further that their magnitude can be bounded solely in terms of d. More precisely, they may be chosen from among the first d+ 5 primes. Letting $$N_i =|Per_{n_i}(\phi)|\geq 1$$ for $$i=1,2,3$$, the action of $\phi$ on the sets of primitive periodic points yields a homomorphism $$Aut(\phi) \to S_{N_1} \times S_{N_2} \times S_{N_3}...................(1)$$ from $Aut(\phi)$ into a product of three symmetric groups. We claim that the homomorphism (1) is injective. To see this, we observe that any f in the kernel of (1) fixes each $Per_{n_i}^{**} (\phi )$; hence f fixes at least three points in $\Bbb P^1(\Bbb C)$; hence f is the identity map. Thus the homomorphism (1) is injective, which implies that $Aut (\phi)$ is finite.

This ends the proof.

Now I have some questions:

1. Can we extend this result for $n=2$. I have some idea here that we don't have finite periodic points in $\Bbb P^2$ but finite indeterminacy locus. Can we exploit that? Can anyone please give me a written answer or some steps to do that?
2. I feel that in $\Bbb P^3$ is infinite, if not, how to prove that? If yes then can anyone give me one example of the rational map $\phi:\Bbb P^3 \to \Bbb P^3$ where $Aut(\phi)$ is infinite.
3. Remember we defined $Aut(\phi)=\{f \in PGL_2(\Bbb C): \phi^f=f^{-1}\phi f(z)=\phi(z)\}$. Now we are going to define $BiRat(\phi)=\{f$ a birational map$: \phi^f=f^{-1}\phi f(z)=\phi(z)\}$. For the sake of simplicity, you can take a birational map to be a rational map with the rational inverse. Then clearly $Aut(\phi) \subseteq BiRat(\phi)$. Now the question is for $n=2$. Is $|BiRat(\phi)|=\infty$? Otherwise give me an example of rational or biration map $\phi$ for which $|BiRat(\phi)|<\infty$?
4. This one is for those who are familiar with the Cremona group. Can we find a finite subgroup of the Cremona group $Cr(2)$ or $Cr(n), n\geq 3$? Give me one specific example.

Thanks in advance. Please help me in at least one of these questions.