You're free to use your purchased logo for any commercial or non-commercial use, and have full ownership of it. The copyright however, applies to your logo as a whole and not the individual elements. We acquire our icons and fonts from a database, so unfortunately we are currently unable to guarantee the copyright of the individual elements, nor do we remove them from our database after purchasing.
We're always available to prove that you designed your logo on Looka if you need evidence in a copyright case, though we do not provide any formal documentation. You can use your purchase dates on your receipt as proof of ownership.
If you're looking to place a trademark on your design, you'll need to do so with the help of an attorney familiar with the laws in your state or country at this time.