I found the answers of the questions $(1)$ and $(2)$ and also an approach for the the third question. Also, I will introduce a software such that it is very good for finding the dimension of arbitrary graphs.
Answer of question $(1)$: The two below graphs$,G_1$ and $G_2,$ are the smallest pairs which are cospectral and also have same dimension.
The spectrum of these graphs are: $\{-1.90321,(-1)^2,0.19394,1,2.70928\},$
The dimension of these two graphs are: $dim(G_1)=dim(G_2)=2$.
Answer of question $(2)$:The two below graphs, left graph is $H_1$ and right graph is $H_2$, are the smallest regular pairs which are cospectral and have different dimension.
The spectrum of these graphs are: $\{-2.56155,(-1)^6,1.56155,3,4\},$
The dimension of these two graphs are: $dim(H_1)=4$ and $dim(H_2)=3$.
I used the program $Dimension-Metrica$ for computing the dimension of graphs. This is a java based program with good GUI. Also, I wrote a $Maple$ program to check the minimality of graphs that is wanted in the question.
For third question, I tried to solve it by $Schwenk's$ method that allow us to construct infinite pairs of cospectral trees. But, as I checked it until now and I believe it is true in general, all these cospectral mates have same dimension. In contrast, I believe the third question is true in general. One approach for solving this question is finding a relation between graph products, such as Cartesian product or strong product, and the dimension of resulting graph based on dimension of its components in product.