Let $\Bbb R^2$ have standard basis $\{e_1,e_2\}$ and $\Bbb R^3$ have standard basis $\{f_1,f_2,f_3\}$.
Then, $T(e_1) = 1f_1+2f_2+0f_3$ and $T(e_2) = 2f_1+(-5)f_2+7(f_3)$.
Therefore, the matrix representation for $T$ is $\begin{bmatrix}1&2\\2&-5\\0&7\end{bmatrix}$.
Let $\Bbb P^n$ have standard basis $\{1,t,t^2,\cdots,t^n\}$.
Then, $T(1) = 0 \cdot 1 + 0 \cdot t + 0 \cdot t^2 + \cdots + 0 \cdot t^{n-1} + 0 \cdot t^n$, and so on.
Also, $T(t^n) = 0 \cdot 1 + 0 \cdot t + 0 \cdot t^2 + \cdots + n \cdot t^{n-1} + 0 \cdot t^n$.
Therefore, the matrix representation is:
$$\begin{bmatrix}
0&1&0&\cdots&0\\
0&0&2&\cdots&0\\
0&0&0&\cdots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\\
0&0&0&\cdots&n\\
0&0&0&\cdots&0
\end{bmatrix}$$
As you already noticed, part (c) of your conjecture is already known, and
the others parts can easily be reduced to it.
Proof of (a) : Suppose that $n \gt m$. The, we have $r \leq m$ and a subpace ${\cal T} \subseteq Hom({\cal V},{\cal W})$ such that ${\mathsf{rank(t}})\leq r$ for every $t\in T$. Let ${\cal W}'$ be a space of dimension $n-m$ and such that ${\cal W}\cap {\cal W}' =\lbrace 0 \rbrace$. You can view $\cal T$ as a subpace of
$Hom({\cal V},{\cal W} \oplus {\cal W}')$, to which part (c) applies.
If we were in the second case of (c), $\cal T$ would be isomorphic to a subpace of $Hom({\cal V}',{\cal W})$ where ${\cal V}'$ is a $r$-dimensional quotient of $\cal V$, whence $\dim({\cal T}) \leq rm \lt rn$, which contradicts the hypothesis.
So we must be in the first case of (c), and there is a $r$-dimensional ${\cal I} \subseteq {\cal W} \oplus {\cal W}'$ such that $\cal T \subseteq Hom({\cal V},I)$ ; we may assume that ${\cal I}\subseteq {\cal W}$ by replacing $\cal I$ with ${\cal I}\cap {\cal W}$, and then we are done.
Proof of (b) : Suppose that $n \lt m$. The, we have $r \leq n$ and a subpace ${\cal T} \subseteq Hom({\cal V},{\cal W})$ such that $\mathsf{rank}(t)\leq r$ for every $t\in T$. Let ${\cal V}'$ be a space of dimension $m-n$ and such that ${\cal V}\cap {\cal V}' =\lbrace 0 \rbrace$. Extending any $t\in\cal T$ by zero on ${\cal V}'$, you can view $\cal T$ as a subpace of $Hom({\cal V} \oplus {\cal V}',{\cal W})$, to which part (c) applies.
If we were in the second case of (c), there would be a $r$-dimensional ${\cal I} \subseteq {\cal W}$ such that $\mathsf{Im}(t)\subseteq {\cal I}$ for any $t\in T$. But then $\cal T$ is isomorphic to a subspace of $Hom({\cal V},{\cal I})$, whence $\dim({\cal T}) \leq rn \lt rm$, which contradicts the hypothesis.
So we must be in the second case of (c), and there is a $r$-dimensional $K \subseteq {\cal V} \oplus {\cal V}'$ such that every $t\in\cal T$ is zero on $\cal K$ ; we can assume that ${\cal K}\supseteq {\cal V}'$ by replacing $\cal K$ with ${\cal K}+ {\cal V}'$, and then we are done.
Best Answer
Intuitively, dimension is the number of degrees of freedom. The elements of $\mathcal{P}_n(\mathbb R)$ are polynomials of degree $n$ (more precisely, at most $n$), so they look like $$ a_0 + a_1 x + a_2 x^2 + \dotsb + a_{n-1} x^{n-1} + a_n x^n $$ To specify such a polynomial, you have to specify $n+1$ numbers, the coefficients $a_0,a_1,\dotsc,a_n$. So there are $n+1$ degrees of freedom in this "space" of polynomials.
To prove that formally, you'd want to think of polynomials $a_0+\dotsb+a_nx^n$ as being linear combinations of the polynomials $1,x,x^2,\dotsc,x^n$, and show that these latter polynomials form a basis. This is done in chapter 2 of Axler.
Again intuitively, a constraint that specifies a single number reduces the number of degrees of freedom by 1. Thus imposing the constraint that we will only work with polynomials $f(x)$ satisfying $f(1)=0$ should, we expect, reduce the dimension from $n+1$ to $n$.
The formal version of this is the rank-nullity theorem (Axler's theorem 3.4), which is why everybody's giving answers involving it. I see Axler doesn't do that until chapter 3, though.
So I think the only thing you can do at this point is to produce an explicit basis for the subspace in question. Exercise 8 in chapter 2 is similar; have you tried that? (And for playing with polynomials, exercises 9 and 12 in the same chapter are good.)
(I have the 2nd edition of Axler's text; hopefully it matches yours.)