If I pay cash will the price be lower
Not likely, but that's a question you will have to ask the seller directly. Most dealers tend to make more through financing, even if it's through a third party loan originator.
Maybe private sales but not dealer sales. Dealers get their money either way. I think Ken is right in that dealers like financing as I suspect they get kick backs. I know a dealer was really pissed at me when I refinanced the day after I bought my truck. They high pressured me to get my loan through them so I fixed them the next day ;)
Used car dealers are notorious for getting an upfront cut of the finance charges. That's why they love pushing high APR loans on unqualified buyers, they're making money even if buyer defaults.