kernel properties/methods and feature maps

[akerman]

I am preparing myself for maths exam and I am really struggling with kernels.
I have following six kernels and I need to prove that each of them is valid and derive feature map.
1) K(x,y) = g(x)g(y), g:R^d -> R
With this one I know it is valid but I don't know how to prove it. Also is g(x) a correct feature map?

2) K(x,y) = x^T * D * y, D is diagonal matrix with no negative entries
With this one I am also sure that it is valid but I have no idea how to prove it or derive feature map

For the following four I don't know anything.
3) K(x,y) = x^T * y - (x^T * y)^2
4) K(x,y) = $\prod_{i=1}^{d} x_{i}y_{i}$∏di=1xiyi
5) cos(angle(x,x'))
6) min(x,x'), x,x' >=0

Please help me as I am very struggling with kernel methods and if you could please provide as much explanation as possible