Can I already try out Kotlin/Wasm or is it not public yet?

It’s public in terms of code, but there are no official public releases

Any links to repo? And possible short intro on build/install and usage, preferrably with IntelliJ?

There is no official intro/usage and probably intellij support too

Is there any ETA for an alpha release? Is it matter of weeks, months, years?

Kotlin team do not announce ETA in general

So when Kotlin WASM will be available, it will not have impact on compose web?

It may be wasm version of compose in future, it's MPP project after all (at least core parts), but I think it's too early for now to tell, even compose web on early stage. It may use the same canvas rendering as js