I want to practice in automating API testing but can't find anything interesting. Everything that I found were on the one hand rather complicated, on the other hand they did not allow testing such methods as POST, PUT and DELETE. Can anybody suggest me some open API which is already online to use as a playground?
Try the petstore - it's a common API used in examples. I used it when working with JMeter and it's a useful API to practice with.
You can do all sorts of different things there.
I think it's a pretty useful site to practice with, and it does seem to be actively maintained - the author has written a couple of courses on API testing which are pretty useful (linked from the first page above).
You may also want to visit this site.
As an example of a simple API from a "real site" is Trello. I've written a blog post showing a CRUD cycle in it for Postman. Nonetheless, it's direct to transcribe the flow to any other automation in testing tool you use.