In addition to Worker's Comp certificates, I would also want to see weekly rosters of covered employees and a roster of who's working on your job each day. Long story, I won't bore with details as to why the list and daily rosters unless someone asks.
Proof of current General Liability coverage is also important. I won't even start on the issues involving proper documentation of legal worker status and what the lack of same can bring on an unsuspecting homeowner should an undocumented worker be injured on the job.
And someone can guarantee their work all day long in writing, but if they're here today and gone tomorrow, that written guarantee isn't worth the paper it's printed on. I think following up on references is the best way, and even asking to see completed and current projects wouldn't be out of line for a sizeable project. Not for just a screen door, but say I'm converting an unfinished attic to a study -- I'd want to talk to at least 2-3 references, and tour the completed projects.
Older, more established firms can be more expensive, but they also have a proven track record and have demonstrated that they are here to stay over the long haul. The new guy in town may have the cheapest bid, but will he be here 3 months from now when that new porch roof is leaking like a sieve and needs warranty repairs?
Anyone doing structural work must be a licensed contractor in the state of Florida, because that sort of work requires a building permit and inspections. All roofing work, even roofing repairs, must be done by a licensed contractor, and new roofing work has to have a permit and inspections. Some types of roofing work must be done by a Florida licensed roofing contractor...
I'm not sure how or why all this got so complicated and byzantine, but it is. Hope this thread helps someone!