Using this parametrisation to transfer the group structure given by multiplication on elements of norm 1 one gets a 1-dimensional rational group law. For q not 0 the law associated to q becomes rationally equivalent to the multiplicative group if coefficients involving the square root of q are allowed. Then indeed the answerm is an emphatic yes. With that understood then we have the following. Two rational group laws are equivalent iff the Lie groups are isomorphic.
Any Lie group corresponds to a rational group law.
The reason can be seen already in the 1-dimensional case where we had to make an extra argument to show that the elements of norm 1 had a rational parametrisation. In the higher dimensional case this isn't even true so that we will have Lie groups which are not given by rational group laws. The answer to question 3 is yes.