WebP does what PNG does, usually at half the size or better. For anything going on a website that is a straightforward win.
WebP supports transparency exactly as PNG does, so logos and cutouts convert without losing their transparent areas. That is the thing people worry about, and it is not a problem.
The saving is substantial. Screenshots and flat graphics commonly drop by half or more, which is why WebP has become standard for websites. For a page with a dozen images it is one of the largest speed improvements available for the least effort.
Support is now universal among current browsers. The only real holdouts are old desktop software, some email clients and the occasional upload form, so keep PNG for anything that has to go into those.
Yes, WebP supports it fully.
It can be either. Lossless WebP still typically beats PNG on size.
No.