The original copyright holder can require the fork be distinguishable and non-confusable with the original project. There is plenty of precedent in that respect.
There isn't much that can be done with lax distribution licenses, but there is still much that can be done under copyright law.
No, trademarks are irrelevant at this point. The original author of a work retains copyright in that work, regardless of the license under which derivatives are distributed. If they have distributed their work under a lax license and then chose to stop distributing under that license, a pre-existing recipient of that code does not suddenly gain copyright. They can not claim the project as their own or lead users to believe they have copyright in the materials they are modifying and distributing other than as a derived work.
For this to come under trademark legislation, there would have to be demonstrable trade. That's hard to show in the case of free distribution.
Actually if it's under a non-restrictive open source licence, it's not fine at all - the code will still be legally available for everyone.