For small businesses, Zenefits and other brokers offer almost exactly the same rates, which are the list rates of the insurance companies. The negotiation only comes in to play with larger companies (currently 50+ employees, next year it will be 100+, I think). For us, Zenefits seemed slightly more expensive, but it is hard to tell since they didn't give us good comparison data based on employee ages and coverage.
If you want to leverage the large company rates as a small company, you need to explore the PEO model, which Zenefits does not support, but ADP does (and several others).
We got cheaper rates with a 'boutique' insurance firm that was smaller. You can get even cheaper by going the PEO route, but we decided that wasn't a route we wanted to go down.
Zenefits does not get any special deals; you get the same deal as every other 'small business' broker.
Nope. In our experience, Zenefits has no special deals and absurdly high pricing for health insurance for small companies. You're far better off going with a PEO (e.g. Trinet etc.)