I think the first bullet is already captured by the fact that the variables are binary or integer.
For the second bullet, you want to minimize the product of length and width, equivalently minimize the sum of logs, which you can linearize as follows. For $k\in\{1,\dots,m\}$, let binary decision variables $\ell_k$ and $w_k$ indicate whether the length or width, respectively, of the grid is $k$. The constraints are:
\begin{align}
\sum_k \ell_k &= 1 \tag1 \\
\sum_k w_k &= 1 \tag2 \\
f_{i_1,j_1} + f_{i_2,j_2} - 1 &\le \sum_{k \ge i_2-i_1} \ell_k &&\text{for $i_1<i_2$ and $(j_1,j_2)\in \{1,\dots,m\}^2$} \tag3 \\
f_{i_1,j_1} + f_{i_2,j_2} - 1 &\le \sum_{k \ge j_2-j_1} w_k &&\text{for $j_1<j_2$ and $(i_1,i_2)\in \{1,\dots,m\}^2$} \tag4
\end{align}
Constraints $(1)$ and $(2)$ enforce one length and one width, respectively. Constraint $(3)$ enforces $$(f_{i_1,j_1} \land f_{i_2,j_2}) \implies \bigvee_{k \ge i_2-i_1} \ell_k.$$ Here, $\land$ is the logical AND operator (true if and only if all arguments are true), and $\bigvee$ is the logical OR operator (true if and only if at least one argument is true). In words, if plantations are placed at $(i_1,j_1)$ and $(i_2,j_2)$, then the length is at least $i_2-i_1$.
Constraint $(4)$ is similar for the width.
The nonlinear objective is to minimize the area $$\left(\sum_{k=1}^m k\ \ell_k\right)\left(\sum_{k=1}^m k\ w_k\right).$$ Because $\log$ is an increasing function, you can equivalently minimize $$\log\left[\left(\sum_{k=1}^m k\ \ell_k\right)\left(\sum_{k=1}^m k\ w_k\right)\right]=\log\left(\sum_{k=1}^m k\ \ell_k\right)+\log\left(\sum_{k=1}^m k\ w_k\right).$$
Because of constraints $(1)$ and $(2)$, this nonlinear function is equal to
the linear function $$\sum_{k=1}^m \log(k)\ell_k+\sum_{k=1}^m \log(k)w_k=\sum_{k=1}^m \log(k) \left(\ell_k + w_k\right).$$
For the third bullet, you can introduce conflict constraints $a_{i_1,j_1} + a_{i_2,j_2} \le 1$ if the distance between $(i_1,j_1)$ and $(i_2,j_2)$ is less than $d$.
