$A = \{(−7, 1), (−5, 1), (−3, 1), (−1, 1), (−7, 3), (−5, 3), (−3, 3), (−1, 3), (−7, 5), (−5, 5), (−3, 5), (−1, 5), (−7, 7), (−5, 7), (−3, 7), (−1, 7)\}$
I tried to plot it and got the following:
I feel like it is incorrect due to the amount of lines. Any help is appreciated.
It is correct.
But there is no reason to draw the lower and the upper elements with that slope.
You don't have, for example, $-7<-5$, under this order.
Notice also this is not really a partial order, since $A$ is not reflexive.
But perhaps you really mean $X=\{-7,-5,-3,-1,1,3,5,7\}$ and $\leq \,= \Delta_X \cup A$, where $$\Delta_X=\{(x,x):x\in X\}.$$ As it is, what you have is a representation of the bipartite graph $K_{4,4}$.