Yes, perhaps I should have asked this separately... But I still don't get the reason for considering (families of) deformations over $k[[t]]$. I still have the feeling that the fibers over $k$-rational points are what I'm interested in. Now, as I learned, when I consider a family over $k[[t]]$ there is only one $k$-point and here the fiber is my original object. So what is interesting about these families? Or what intuition should I have here? (In case of $k[t]$ I think I have a good (the correct) intuition...

Ah, of course! Thanks. Can you also give me a hint concerning my deformation question above?

Why is the kernel maximal? (This is probably obvious...)

Okay, a question into another direction: Why do people consider deformations parametrized by $k[[t]]$? Parametrizing over $k[t]$ makes perfect sense to me; I have a fiber over any $\alpha \in k$. But in case of $k[[t]]$?