Well first of all it all is dependent on your project you are trying to get published. Your right, it shouldn't take them 2 weeks to come up with something like that unless your talking about EA or Ubisoft. I've never heard of 'Funvertising' before, so I can't tell you I'm familiar with what they do (but I can assume from the title itself).

However, I think you should have sent in at least a demo. Before any publisher will sign your game they will want to see (or should want to see) some of the following materials:
- Playable Demo
- Design Document
- Screenshots
- Videos
- Etc.
BUT, Do not release anything prior to signing at least an NDA (Non-Disclosure Agreement). This is a good way to protect your material of the game from them, and if you want to take it a step further then get a Copyright on your idea/art/etc. depending on how serious you are with your project. Releasing a demo or some sort of additional media to them will give them a firm idea of what they are working with.

I will say this, dont expect cash before sending them a demo. They might laugh.