Does the product of two invertible matrix remain invertible?

Of course: $B$ invertible implies $B^T$ invertible, and the product of two invertible matrices is clearly invertible.

This is easily seen from these equations: $$BB^{-1}=I\implies (BB^{-1})^T=I\implies (B^{-1})^TB^T=1,$$ and the fact that if $X$ and $Y$ are invertible, $(XY)^{-1}=Y^{-1}X^{-1}$.

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Perhaps the general properties you should take away are these:

$(XY)^T=Y^TX^T$ and $(XY)^{-1}=Y^{-1}X^{-1}$.

Yes. $$ \det(B^T\,A)=\det(B^T)\det(A)=\det(B)\det(A)\ne0. $$ Moreover $$ (B^T\,A)^{-1}=A^{-1}(B^{-1})^T. $$