The first two sentences have basically the same meaning. Keith is romantically interested in Jennifer, but he doesn't know if she feels the same way, so he is hesitant to ask her out on a date.
In the third sentence, "make a date" is correct usage, but it implies either a business or friendly "let's get together and find out what you've been up to" appointment, not a romantic outing. So it's not likely that Keith would worry about being turned down.
"Have a date" is not typically used in AmE.