Double

To take it from the beginning, you are correct about mathematical laws. The only thing I would add is that mathematical laws are relational and depend on the meaning of the terms within the equations. We consider the laws to be necessary, but the necessity of the laws already assume that numbers and mathematical operators have set values. If we find an alien civilization, it is possible that they use different notation than us, so instead of counting [1,2,3,4], they might count [7,2,0,3]. In these cases, the actual symbols are not important. It is the meaning of the symbols that are important. The main point is that 1 plus 1 could equal 3, but it would require that the meaning of either 1 or 3 be changed (or the meaning of the plus or equals sign- I think you get the point).

As to the first point, physical laws do depend on mathematics. Physical laws are discovered by humans, and the laws of physics in particular are articulated in terms of mathematical language. If mathematics was not true, the laws of physics (e.g. the inverse-square law of gravity) would be non-sense. As to whether the physical laws themselves actually depend on mathematics, I would still argue they do, but that question might be more open to debate. However, I think most physicists would conceed that physics depends on mathematics. The main point is that the actual act of discovering and explaining the laws of physics will always depend on mathematics (as well as observation, hence, the physical world).