Hello
I am new too K2 so I don't know much. When I add a K2 category, the category seems to be added as a trashed category view as welll as the current category view but I can't delete it from the trash as there is no tick box beside it. Is this normal?

Yes perfectly normal, trashed categories are the ones that can be selected and erased.

I am facing the same problem but in my case the same categories (the have the same id) appear both as current and trashed. In the "trashed" category I get only one check box at the top of the tree, by the word "title". In other words, I cannot delete the "trashed" categories as, in fact, they are the "current" categories appearing in the trashed category for some mysterious reason.
Any ideas, please?

The categories cannot share the same id.
Have you used a 3rd party extension like Additional Categories for K2?