Can a group be union of three proper subgroups? What about vector spaces?

Answer 1

Yes in both cases! (although for vector spaces, it depends whether you're allowed to pick which field you're working over. It's not true for vector spaces over the real numbers, for example)

Here's a hint: think about the smallest group there is that isn't cyclic.

Once you figure it out for groups, you'll realize the same answer works as a vector space over an appropriate (small!) field.

Answer 2

Yes