I don't know if CSS is Turing complete already, but I'm pretty sure it would have to be if it allowed you to implement something like that...
No, it would only need to be primitive recursive [0] because you can bound its runtime. There's actually a lot you can do without Turing completeness.
[0]: https://en.wikipedia.org/wiki/Primitive_recursive_function#C...
Pure CSS game: https://codepen.io/elad2412/pen/hBaqo
Rule110 (After setting first row requires alternation of tab/space to run): https://codepen.io/elrumordelaluz/pen/wqLyH?editors=010
stackoverflow discussion: https://stackoverflow.com/questions/2497146/is-css-turing-co...
CSS4, let's do it.