Are there drivers who do both taxi and Uber/Lyft on the same shift? Taxi drivers lease their cabs, and Uber drivers can rent their cars. I assume some people do both?

In order to drive for Uber and Lyft, you have to register your personal car with them. They take note of car's model, make, year, license plate, etc. With leased taxicabs, you wouldn't be able to provide these. They are supposed to be personal vehicles and must be approved.

There are rental services with rideshares, but they have official relationships with Uber and Lyft. There's usually even a different payment plan where the costs of vehicles are deducted. You can't just rent a car on your own (or even borrow one from someone) and expect to be able to drive.

Also, if you are in a big city, one of the biggest advantages of driving a taxi is being able to be "hailed" by people on the curbs, legally. They can raise their hands and you can pick them up. Believe it or not, that requires special permits (usually shown in the form of a medallion). It doesn't hurt to supplement the workload by also picking up Uber/Lyft customers, but that shouldn't be your primary target. again in big cities.

How fun would that be though. 

Taxicabs are not allowed by Uber and Lyft, so no. If you call for an Uber and a taxi shows up, they are breaking the rules.