"Raw" latex consists of long hydrocarbon chains. The chains themselves have covalent bonds and are therefore very strong; imagine them like hemp threads. A volume of latex is basically a huge mess ("ball of wool") of those chains, which are weakly held toghether by van der waals forces.
This raw latex is therefore a thermoplast: It is soft and sticky when hot and rather brittle when cold. This must have been rather annoying when only this form was known. It is also not very elastic but malleable, (is that the word?)
But now it was discovered that sulfur likes to form cross-connections between said chains. This basically makes the latex a single, huge molecule (hence a duroplast). But there are not too many of those cross connections, and the material looks rather like a loose net. If this vulcanized latex is streched, it can regain its form rather well; it is therefore called an elastomer. The more sulfur-bridges there are, the harder is the material; i.e. a hair band has less bridges than an ice hockey puck.
I think this view should explain the behaviour of latex quite well and on an intuitive level.
Testing for crosslinks seems rather difficult in general; but in the case of latex you can test for sulfur however. If you only have an emulsion this is very easy, if it smells repugnant there is sulfur in it. If you have solid material you might as well just test for elasticity.
In general it seems a save bet that elastic materials have crosslinks.