Bike advice

I vote for the used Fastback, if everything on it works as it should and it fits reasonably well. It's from a period when Schwinn tried to address two markets at the same time, independent bike dealers and big-box discount stores. This was one of their IBD bikes, quite nice. And 10-speed 105 (5600) is a solid workhorse of a drivetrain.

The Giordano Libero is not a 24" men's bike, it's a kids' bike with 24" wheels. That should close the debate right there, unless you happen to be shorter than 4'10". The Schwinn Phocus is another cheap bike made for big-box stores. It will be built, but not well. Figure on $70-80 to make it ready for your charity ride. Big-box stores don't service the bikes they sell. They give you store credit if it's inside 30 days and you have a receipt, and throw the bike into the crusher.

If anything is out of adjustment, though, don't hesitate to downgrade your offer and then have the bike tuned by a good bike shop. Fiddling with derailleur cables is not something a newbie should be doing days before a charity ride.

I second the vote for the used Schwinn as long as it wasn't abused and it looks obvious it was taken care of or at least rarely ridden. The 105 components on the Schwinn are the most reliable groupset that Shimano makes and works very smoothly.