First question
A Gaussian process needs a fixed domain of definition, and every observed data point as well as every prediction point has to be in that domain. So, no, you cannot have "partially qualified" points as inputs.
That said, there are workarounds. Whether these make sense or not depends on your use-case and the probabilistic structure you would like to model. I will give two examples below.
Example 1:
The points where additional measurements are available are fixed and known in advance. Say $x\in\mathbb{R}$ and you have additional measurements $z\in\mathbb{R}$ if, and only if, $x=0$. Your domain of definition is then the "cross" $\mathscr{C}=\{(x,z)\lvert x=0 \text{ or } z=0.\}$ Define a covariance function on $\mathscr{C}$ and you are in business. With this setup you cannot predict values at locations not in $\mathscr{C}$ such as $(1,1)$ or use such points as input.
Example 2:
If you do not know the points in advance, you can combine two Gaussian processes. One defined on $x$ values, i.e. $G_1(x)$ and one defined on $z$-Values, $G_2(z)$. The way those are combined is again determined by the covariance. Two popular choices are tensor product or direct sum, where the joint covariance function is either the product or the sum of the covariance of $G_1$ and $G_2$.
Second question
Additional information will reduce uncertainty. The reduction will be stronger the "closer" points of prediction are to points of observation, be those $Z$ or $X$ related ones. How big the reduction is and what "closer" specifically means, is solely determined by your choice of correlation function. It will vary widely depending on this choice.
