I don't know whether this is useful in practice, but the straightforward generalization of your 0-connectedness criterion to a 1-connectedness criterion goes as follows.
Any 1-cycle of the form $[(a_0, a_1), (a_1, a_2), \ldots, (a_k, a_0)]$ can be transformed into the trivial (empty) 1-cycle by a sequence of the following moves:
$[\ldots, (a, b), (b,c),\ldots] \to [\ldots, (a,c),\ldots]$ if $a$, $b$ and $c$ are mutually disjoint (i.e. they correspond to a 2-cell in the complex)
$[\ldots, (a,c),\ldots] \to [\ldots, (a, b), (b,c),\ldots]$ if $a$, $b$ and $c$ are mutually disjoint
$[\ldots, (a, b), (b,b'),\ldots] \to [\ldots, (a, b),\ldots]$ if $b$ and $b'$ are parallel (remove degenerate 1-cell)
$[\ldots, (a, b), \ldots] \to [\ldots, (a, b),(b,b'),\ldots]$ if $b$ and $b'$ are parallel (insert degenerate 1-cell)
$[(a, b), (b,c), (c, a)] \to [\;\;]$ (empty) if $a$, $b$ and $c$ are mutually disjoint
The basic idea is that a triangulated disk is "shellable" in some sense. There should be a similar criterion for general $k$-connectedness (in terms of $k$-dimensional Pachner moves on triangulations of the $k$-sphere), but I haven't thought carefully about the shellability issues in higher dimensions.
EDIT:
I think that allowing degenerate simplices (i.e. parallel disjoint curves) means that one doesn't need to worry about shellability.
In case it wasn't clear what I meant by Pachner moves, here they are for $k=2$. The context is a triangulation of the 2-sphere where the vertices are labeled by curves and the curves at the vertices of a $j$-simplex ($j=1,2$) are mutually disjoint.
Two adjacent triangles $(a,b,c),(b,c,d)$ can be replaced by $(a,d,b), (a,d,c)$ if $a$ and $d$ are disjoint.
A single triangle $(a,b,c)$ can be replaced by three adjacent triangles $(a,b,d),(a,c,d),(b,c,d)$ if $d$ is disjoint from $a$, $b$ and $c$ (and we can also do the reverse move, replacing three triangles by one).
