More Generally.
First you are going to want to set this matrix up as an Augmented Matrix
where $Ax=0$.
$1)$ To find the rank, simply put the Matrix in REF or RREF
$\left[\begin{array}{ccc|c}
0 & 0 & 0 &0 \\
0 & 0.5 & -0.5 & 0 \\
0 & -0.5 & 0.5 & 0 \end{array}\right] \longrightarrow RREF \longrightarrow \left[\begin{array}{ccc|c}
0 & 0 & 0 &0\\
0 & 0.5 & -0.5 & 0\\
0 & 0 & 0 & 0 \end{array}\right] $
Seeing that we only have one leading variable we can now say that the rank is 1.
$2)$ To find nullity of the matrix simply subtract the rank of our Matrix from the total number of columns.
So:
Null (A)=3 - 1=2
Hope this is helpful.
Let's go back to definitions. Range is a set-theoretic concept. Given a set map $f:X \to Y$, we denote the image or range as the set of values in $Y$, i.e. $\{y\in Y|y=f(x) \text{ for some }x\in X\}$.
Your other concepts live in the world of linear algebra. Let me define things abstractly, rather than by matrices. Given a linear transformation $T:V\to W$, we define the nullspace to the set of vectors $v\in V$ such that $T(v)=0$, i.e. $T^-(0)$. It turns out both the range and nullspace of a transformation $T$ happen to be subspaces in their appropriate domains. Since dimension is a well defined concept (over fields), we can ask for the dimensions of these two subspaces. The dimension of the range is defined to be the rank, while the dimension of the nullspace is defined to be the nullity.
How does all this relate to matrices? Recall the data of a matrix of a transformation $T$ is that the columns of the matrix denote where a basis of $V$ lands after applying $T$. Extending the transformation linearly (i.e. You know what $T(e_1),...T(e_n)$ is, so $T(a_1e_1+...+a_ne_n):=a_1T(e_1)+...+a_nT(e_n)$), uniquely(linear independence of e_i!) extends (e_i spans V!) $T$ to the entirety of $V$.
Knowing this, we can start translating all our definitions from the abstract into statements about matrices. For example, we see the range of a matrix is the Span of the columns. The rank of a matrix would then the the number of linearly independent columns. For example, the 2x2 matrix consisting of all 1's will have rank 1 despite being a transformation two 2-dimensional spaces.
How would we define nullspace of a matrix? Denoting the matrix by $M$, it would be the subspace of vectors $v\in V$ such that $Mv=0$. Concretely, this means for you to solve a system of equations, and is what linear algebra was built to do. Now to define nullity of a matrix, we can use the rank-nullity theorem which tells us $\dim(V)=rk(T)+nul(T)$, so we can define nullity of the matrix as $\dim(V)-rk(T)$.
Some conceptual mistakes I saw in your post: you're confusing nullity with nullspace. The former is a natural number, while the latter is a subspace of V, NOT in general a vector (unless the kernel is just 0). The phrase the rank of the kernel makes no sense; it only makes sense to talk about the rank of a transformation. Similar, the nullity of the rank also makes no sense.
Best Answer
$L$ is onto$\implies Im(L)=\mathbb R^7\implies \text{rank}(L)=7.$
$\text{nullity}(L)=\dim M_{4\times 3}(\mathbb R)-7=12-7=5.$