Can we register a free icon as a trademark?

By virtue of your commercializing the app and the mark that identifies it, you have already "used" the mark in U.S. Commerce (assuming the downloads were accessible by U.S. citizens) and acquired rights in the mark if there are no pre-existing rights belonging to anyone else. Registering the mark in the U.S. would merely publicize your use and claim of rights in the mark whether or not that claim to the mark is allowed.
Whether you are allowed to claim rights in the mark and use it commercially would be dictated by the terms of the Creative Commons license. Does that license allow you to appropriate IP for commercial use? I'm not sure. If it does, and you can permanently acquire those rights for commercial use, then the answer to your question should be yes. If the answer is no, then you may already have violated the CCL.
Now whether a registration would issue from the U.S. Patent and Trademark Office under those circumstances is more a question of logistics and USPTO operations than legal rights. What I mean is that the USPTO might register your mark if there is no conflicting mark on its register because the USPTO won't look beyond its register; but the administrator of the CCL or whomever owns/owned the design prior to the CCL that might still claim rights in it could oppose your registration of the mark and the validity of that registration could always be challenged based upon whether you had the right to register a mark from the CCL.

You can only register a logo if you own the right of reproduction of it.

By registering the logo, you will not acquire the exclusive right on it, as long as it was not linked to your image before.

In clear, this is a bad idea, as usually free logo, are not free for all utilization you can do with, but only through a license you have right to use it for some use.

Free logo or public domain remain limited to private use and you cannot build your communication and your company image on something that can be used by other for something else and if you cannot track clearly the owner of the free logo, you could face litigation for infringement.

Best way is to keep the general idea of the logo, and then to pay a designer to do an original one, to have a transfer of the intellectual right for commercial use and then to be able to sleep in peace.