Petrol scooters (motor scooters for us Yanks) are a wonderful hobby and mode of transportation. I've said it a dozen times on this and other blogs, but my favorite scooter line is the 125cc scooters for on-road use, but I generally like something with a bit more torque and power for offroad use.

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!