Is it possible to insert product attribute values using Web API out of the box (ver 4.60)?
If yes - can anybody please provide sample code?
I haven't found any CUD implementation in Web API sources for ProductAttributeMappingDto and ProductAttributeValueDto so far.

Out of the box you are able to make following requests related with product attributes: https://docs.grandnode.com/operations-with-product-attributes. In the case of other actions, you can extend the API via plugin. Check the following topic.