Can I use in my portfolio work that I made for competitions and rejected items (logos, brochures, etc.), which include the brands of the companies organizing these competitions?
Unless there is a written contract (which you signed) you OWN the work product. Additionally you have certain 'moral rights' to use the work in your portfolio.
As to rejected work. That's cearly the property of the artist or their studio.
Our studio has staff designers who work under 'work for hire' agreements. Even so our contacts allow (encourage) these designers to include their work product for our studio in their portfolio with approprate copyright and attributes.
This policy applies to interns as well. I would add interns are paid (as they should be) fairly for their efforts.
This is totally up to you, assuming the terms of these competitions/etc allow you to use the work for self promotion.
If you look at diffent portfolios or browse through the Behance showcase, you will see some designers will also include rejected material, proposals, contest entries, etc. It will sometimes be relevant to potential new clients to present these, but be advised some other clients are more interested in seeing work that actually got produced.