gn.general topology - A question about indecomposable continua.

As pointed out by Jeff, the notion you define may not really be what you are after, since indecomposable continua are not 'indecomposable' in your sense. However, we can ask:

Is there a nontrivial connected metric space $X$ such that $X$ cannot be written as the union of two proper connected subsets?

The answer, as Jeff suggested, is no.

Indeed, let $X$ be a nontrivial connected metric space. If $X$ does not have any cut-points, then clearly we can write
$$X = (xsetminus{x_0}) cup (Xsetminus{x_1})$$
for some $x_0neq x_1$, and are done.

If $X$ does have a cut-point $x_0$, let $A$ and $B$ be open subsets of $X$ such that
$$Acap B = {x_0}; quad Asetminus{x_0},Bsetminus{x_0}neqemptyset quadtext{and}quad Acup B = X.$$

We claim that $A$ and $B$ are connected. Indeed, if $Uni x_0$ is relatively open and closed in $A$, then $Ucup B$ is open and closed in $X$, so we must have $U=A$ (since $X$ is connected).

Regarding your question on the number of proper connected subsets, we can still ask the following question:

If $X$ is any nontrivial connected metric space, what can be said about the cardinality of the set $S$ of proper connected subsets of $X$?

It seems plausible that the set $S$ has at least the cardinality of the continuum, but I wasn't able to find a reference (and haven't thought very deeply about it). Certainly the set $S$ must be infinite.