This is a rapidly evolving area. Here's the current state:
Copyright: The purely AI-generated portions are not copyrightable (per Thaler v. Perlmutter and the Copyright Office's guidance on AI works). BUT — your substantial human modifications likely create a copyrightable derivative work. The Copyright Office has registered works with AI assistance where the human contribution was substantial (see the Zarya of the Dawn registration).
Trademark: Trademark registration doesn't require copyright ownership. It requires that the mark is distinctive and used in commerce. You can trademark an AI-assisted logo. The PTO looks at distinctiveness, not authorship origin.
For the client relationship: your contract should address IP ownership. If your freelance agreement assigns all work product to the client, the copyrightable portions (your modifications) transfer to them.