I might be missing something about question 3. Here's a simple construction.:
Consider a projective space $P$ of dimension $\text{dim}\, P_1 + \text{dim}\, P_2$ that contains both $P_1$ and $P_2$ in general position. Then each point of $P-P_1-P_2\ $ lies on exactly one line connecting $x$ from $P_1$ with $y\in P_2$. Is this the kind of join you're looking for?
About question 2, I have a simpler thing that isn't clear to me (now posted as a questionfirst):
is a complex algebraic variety which is topologically contractible necessarily an affine line? If not, is it necessarily affine?
