Can car manufacturers put their brand name on the COT? who puts engines in the COT?

May be...

Manufacturers are providing the engines & parts, the individual teams put them together.

The COT - the car itself is generic. But they put the Chevy, Ford, Dodge, or Toyota logo on the front depending on what engine is under the hood.

Yeah, it's a way to get publicity for there company. The Car manufactures give them the engine, but the parts are the teams problem.

Ford, Chevrolet and Dodge provide engine blocks and bare cylinder heads. Toyota's american racing arm TRD provides complete engines.

The cars have slightly different front noses depending on the manufacturer. Ford, Chevrolet and Dodge do not build any cars. Toyota (TRD) has built some complete cars.