Will it cause any problem if I use clips from public domain cartoons on a commercial website?

Given that copyright laws extend back to approximately the 1920's now, I doubt there is much available in the public domain yet. If the material is in the public domain, you may be able to use it and make the adaptations that you describe. But be aware that there may be other kinds of protection on the work that would prevent you from doing this. Check with an IP attorney to see if what you have in mind is feasible.

YES - it will likely cause a problem. First - how do you know the clips you want to use are "public domain" Copyright protection lasts a LONG time. Changes made may be viewed as a derivative work - which only the copyright owner is entitled to do. To be safe - create your own work from scratch.