1. Many bidet taps are based on wash basin taps but with a swivelling end to the spout (with limited movement).
2. Search for "bidet mixer" taps on the internet.
3. If by "pop up" you mean the sort with a rod at the back of the tap to open, I find them generally OK, if a bit of a nuisance to fit.
4. The "click clack" type, where you push once to close and again to open are, in my experience, far more prone to failure. Some people call these pop up plugs.