Ooh Squibs! I love Squibs.
1. Squibs are absolutely playable characters on the board. We have one or two (although I think they're largely inactive now) on the Elsewhere boards. Because Squibs have no magical ability - they're essentially Wizard-born muggles - they generally aren't admitted to Hogwarts, but the Squibs who have been created and played so far are still able to reside in the magical world, a la Argus Filch from canon.
2. In general, Squibs would be subject to the same laws and regulations and restrictions as Muggleborn wizards. That said, some establishments may be willing to employ them in roles that don't require much magic (such a receptionist or something), and they aren't exempt from treatment at Mungo's. However, as most careers in the Wizarding world require at least some magic to perform the required job functions, I expect it would be incredibly hard for a Squib to find employment.
That doesn't mean, of course, that a Squib couldn't employ himself in...whatever manner or legal or illegal means are available to him. (But you didn't hear that from me.
3. I think I've pretty much covered this above, but as Squibs have magical blood, they are able to view and interact with the magical world. Think again of Argus Filch or Arabella Figg from HP canon - Mrs. Figg, in particular, was quite helpful to Dumbledore. Their restrictions are mainly self-imposed, as they can't perform their own magic, and Squibs need to be transported through assisted means, like Side-Along Apparition.
We don't actually have an 'S' badge (although maybe we should!) because there is so little call for Squibs. Instead, we put it in your subbie - where students display their year and levels, Squibs just say 'Squib'.
Hope that was helpful! Please let me know if you have other questions, or if I missed anything.