Yes! You become the complete owner of your purchased logo and can use wherever you want. But this applies to your logo as a whole and not on any particular element. Elements such as icons and fonts are acquired from a database so we cannot guarantee the copyright of the individual elements, nor do we remove them from our database after purchasing. We also do not assure that the elements used in your logo will not be used in other logo. So, you have full ownership of your logo as a whole and not on any elements used in it.
For the evidence of your copyright, you will have purchase date on your receipt. Designhill is always with you to prove the ownership if required.
If you want to place trademark in your logo, you need to seek help of an attorney having knowledge of current laws in your country of residence. This is entirely a case of your local laws and Designhill cannot help you out in this matter.
For additional information please read our article on trademarking, and our Terms of Service.