Is it true that $\left(\phi^\dagger \phi\right)^2$ is invariant under $U\left(1\right) \otimes SU\left(2\right)$ where $\phi$ is the Higgs field $(1,2,1/2)$?

Yes, clearly. The Higgs field $\phi $ is invariant under the transformation mentioned. To quote wikipedia:

In the standard model, the Higgs field is an $SU(2)$ doublet, a complex spinor with four real components (or equivalently with two complex components). Its weak hypercharge ($U(1)$ charge) is 1. That means that it transforms as a spinor under $SU(2)$. Under $U(1)$ rotations, it is multiplied by a phase, which thus mixes the real and imaginary parts of the complex spinor into each other—so this is ''not the same'' as two complex spinors mixing under $U(1)$ (which would have eight real components between them), but instead is the spinor representation of the group $U(2)$ .

And the conclusion follows.

Does this invariance imply that its hypercharge is invariant under $U\left(1\right)$ and its spin is invariant under $SU\left(2\right)$? .

Yup, see the quote from wikipedia above-mentioned. Almost by' definition.

Make separate questions for the rest.

P.S. You may find this link useful for learning QFT. It is more of QCD, but it covers other things too.