Good questions. First, surety bonds are not generally required on privately owned projects. They are allowed, and a property owner and/or a prime contractor could require them for a private project, but they are not "required" in the same sense that payment bonds are required on public projects. For further talk on Florida payment bonds on private projects, this is a great resource: Conditional Payment Bonds on Florida Private Projects
Second, a Florida mechanics lien can be enforced (or "foreclosed") anytime after filing, but up to before 1 year from the lien filing date. The timeframe can be extended or shortened, though. If more work is done after the lien was filed, and if that work is unpaid and requires another lien claim for payment (via amending the originally filed lien), then the deadline will run from the time that the amendment is filed. As for shortening the timeframe - an owner can do this by contesting the lien. If contested via a Notice of Contest, the time for filing a lien enforcement suit is 60 days from when the lien is contested. But, if the lien is contested via the owner or GC serving a summons and complaint, that lien enforcement deadline is shortened to a mere 20 days from when it's contested.
As a final note, keep in mind that when a mechanics lien has been filed, adding a step before the foreclosure process can really help to speed up payment and potentially save a claimant from having to proceed with the expensive and time consuming foreclosure process. By sending a document like a Notice of Intent to Foreclose, lien claimant can give their customer and/or the owner one last chance to make payment before a lawsuit is filed to enforce the lien. Plus, if it's ineffective, a claimant can always proceed with their suit.
I hope that was helpful information! For more information on Florida's lien laws, this is an excellent resource: Florida Mechanics Lien Overview.