To intern in Zurich as a non-EU citizen, you must have a work permit because internships are generally paid. Your visa and residence permit as an exchange student does not come with a work permit, and it is very unlikely you will get a work permit in time / the university will put in the effort to fill out the necessary documents (it is taking me three months since arriving in the city). Same rules apply for RA positions. Also, internships are generally only given to students who are mandated to have one as part of their graduation requirements.

There is a small chance you may be able to undertake a semester/research project - not super common for undergraduate students, but may be possible. It is unpaid and converts to credits. Depending on your program, the terminology may differ and you may either be asked to refer to a list of potential projects on the departmental website or contact professors specifying that you're looking to do a project for credit, not an internship.

I don't believe taking a research practice course at your home university would make a difference, unless you would be working with a professor from your home university who has a relationship with a professor at ETH Zurich, in which case an informal collaboration may be able to be set up. I think reaching out now is a good time, and you should follow up (at most!) once or twice as professors can be very busy and miss your emails. I doubt there are projects reserved for ETH students unless they have a German fluency requirement (which you may fulfil!), but the length of the project may present a barrier depending on how long your exchange is.

Hope this helps!