Yes, we can solve (not "prove") combinatorial problems by means of probabilities; this is the probabilistic method. However, while I'm not sure I understand what you mean by "the probability of occurrence of a rectangle in any $4$ selection is $1$" (what is "any $4$ selection"?), it doesn't sound like a correct application of the probabilistic method to me (as far as I understand it).
There are $\binom72=21$ different pairs of $x$ coordinates. Placing $k_i$ points in row $i$ creates $k_i(k_i-1)/2$ different pairs. If the same pair is created in two different rows, we have a rectangle. With $\sum_i k_i=22$, we have
pairs in total. Since $\sum_ik_i$ is fixed, $\sum_ik_i^2$ is least if the points are distributed over the rows as uniformly as possible, i.e. with six $k_i$ equal to $3$ and one equal to $4$. Then
so at least one pair must occur at least twice, so there must be a rectangle.
This is an application not of the probabilistic method but of the pigeonhole principle. However, you can view the pigeonhole principle as a special case of the probabilistic method: In a typical application of the probabilistic method, one shows that the average of an integer is less than $1$, and it follows that there is at least one instance where it is $0$. In applying the pigeonhole principle, one shows that the average of an integer is greater than $1$, and it follows that there is at least one instance where it is at least $2$.