Absolutely not, no. At least not any more than LaTeX is a cloud based solution with monthly fees (Overleaf)

The language, ecosystem, and compiler are FOSS. There is a cloud editor / collaboration platform that is paid, but nothing about the language requires that you use it (I use it almost exclusively through emacs)

No, Typst the typesetting software is FOSS (Apache license), as is a bunch of the surrounding ecosystem (e.g. the LSP for editor support). The people making it also ofter paid SASS stuff for features enterprises like, but there is no need to use them.

> Typst seems to be a cloud based solution with monthly fees?

In the exact same way that git is: it isn’t, but one easy way to use it is.