The final answer is $1403$

My solution during the exam: Let $r_i$ and $c_j$ be $i$-th row and $j$-th column respectively. Let $T$ be the best score that the second player can achieve, no matter how the first player acts. Claim 1. $T\ge 1403$ Proof. Let the first player choose $r_i$ at one of his moves, if the second player selects $c_i$ and the second player will do the same strategy again and again, it's clear that the cells with coordinate $i,i$ ($1\le i\le 1403$) will be red. Now note that among cells $(i,j)$ and $(j,i)$, one is red, thus the number of red cells is $\ge 1403\times 702$, so $T\ge 1403\times 702 - 1403\times 701=1403$ Claim 2. $T\le 1403$ Proof. Here is the strategy for first player: Each time, he will choose the largest amount of $k$ such that $r_1,\dots,r_{k-1}, c_1,\dots,c_{k-1}$ are already chosen, and then he selects $r_k$ if it isn't selected, else he will choose $c_k$. It's clear that this strategy works perfectly since the cell $(i,j)$ , $i<j$ is red, if $c_j$ is chosen by the second player and the cell $(i,j)$ , $i>j$ is red, if $r_i$ is chosen by the second player, thus the number of red cells at the end is$\le 1+2+\dots +1403=1403\times 702$, so $T\le 1403$ and we are done!

Gholam can always guarantee a score of $\boxed{1403}$. Gholam's strategy to ensure he gets to a score of 1403: If Sahand changes a coulmn then Gholam should change a row and if Sahand changes a row then Gholam should change a column. We will show that every set of turns Gholam's score increases by exactly $1$. Let Sahand and Gholam have previously turned over $S_R$, $S_C$, and $G_R$, $G_C$ rows and coulmns respectively. Notice $S_R=G_C$ and $S_C=G_R$. WLOG assume that Sahand chooses to turn over a row next. Then Sahnd's score will increase by $n-S_C+G_C$. On his turn, Gholam's score will increase by $n-G_R+S_R+1$. Then Gholam gains a point each round. Sahand's strategy to ensure he gets to a score of -1403: After the first move Sahand should always play perpendicular of Gholam. Using similar reasoning each round Sahand's score will not decrease. Considering that Sahand's score was at $1403$ after the first move then after $2805$ moves he will still be able to guarantee a score of $1403$. Gholam's final move will move his score down to at worst $-1403$. Combined these finish the question.

Any solution for the Sahand's strategy with induction?