Which tool would you recommend to create high professional animations for websites (not mobile Apps)?
CSS + JS.
Velocity.js, Famo.us, TweeneJS, Three.js, Two.js, ...
Velocity.js but stick to adding/removing classes and CSS transition where you can. The trick for good looking animation is timing and easing/spring.
Sounds like https://easee.design/ would be the right tool for you. CSS + JS work as well, like others said.