About questions 3 and 4:

3. It's known that for any countable ordinal $\alpha$ there is a subset of the reals with order type $\alpha$ under the usual order. Using this, given a countable ordinal $\alpha$ and a sequence of rationals $(x_\beta)_{\beta<\alpha}$ such that $((x_\beta)_{\beta<\alpha},<)$ has order type $\alpha$, if we set $z_\beta=\begin{cases}x_{\beta+1}-x_\beta\textrm{ if }\beta+1\le\alpha \\ \lim_{\gamma\rightarrow\beta}z_\gamma\textrm{ otherwise}\end{cases}$ for all $\beta<\alpha$, we have a convergent series of reals of length $\alpha$. (We can show the series converges to a real using transfinite induction up to $\alpha$ on $\beta$, at limit steps using the fact that sums commute with limits.)
4. The definition of sums for length-$\alpha$ series should work for any ordinal $\alpha$, countable or not, by transfinite recursion.