A style of scooter that is increasing in popular are the Schwinn electric scooters. While we don't cover bike and sports scooters much here, you can find dirt cheap Schwinn electric scooters via our partner outlet store or out sister bike-scooters.com blog.

I'm wondering if I can buy a 50cc or below motor scooter and use it in the bikers lane or sidewalk (i.e. off the street) without being 16. Also, I live in Texas.

you won't find out here...pick up the telephone and call your local area DMV (Department of Motor Vehicles) because that scooter IS a motor vehicle and therefore is under the jurisdiction of the cops, etc.
So give them a call!
I know in our state (NH) minors (under 16) cannot legally drive ANY vehicle with a motor-scooters, atv's, snowmobiles, etc. off their OWN private property-doesn't matter if it's in the backwoods or not-they are not supposed to go off their own land without permission of the other landowners, AND a legal adult with them. Of course, everyone ignores it-but when there's an accident or complaint-the parents get a ticket and the snowmobile or atv can be confiscated!
So find out before you buy it!
To drive ANY vehicle of any type...is a PRIVILEGE, and NOT a 'right'. So there is nothing that 'guarantees' you the right to do so!