You are correct in your impressions of NSF. Their RFPs tend to be the most broad because the agency's mission isn't limited to a specific scope (like USDA, DoE, or DoD). They do also tend to be the most competitive, though, because they usually get the most submissions.
As a whole, NSF also tends to be the most open/inclusive of all the funding agencies. Meaning getting funding isn't as dependent on "who you know" as it is with DoD or DoE. I'd encourage you to call the program manager to ask how many proposals they are typically getting per RFP and what percentage are being funded. I would also use their website to search and see exactly what types of projects are being funded (they have a public database called Fastlane that is accessible through their website). With that said, the agencies I worked with re small business/university collaborations were USDA and DoE, and it has been several years since I worked with any funding agency.
As far as applying goes, you need to leave PLENTY of time. Universities have offices of sponsored research that have to review every single research proposal that includes university resources (even if you/your business will be the one submitting the proposal). The staff members in this office DO NOT like to do things last minute. I'm fact, the university I worked for had a policy that all proposals had to be submitted for review at least one or two weeks (can't remember exactly which because it's been so long) before the deadline. There is enough red tape in the university approval process that, if you don't plan accordingly, you could miss the application deadline.