I would like to know if the name of an open-source application can be trademarked by the original author and what it means when it comes to distributing it through major linux distros, that is, will they accept this kind of software?
closed as off topic by Gary Rowe, Thomas Owens♦ Apr 23 '12 at 21:47
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Open source organizations certainly trademark their names and the software names. Take a look at the Mozilla Trademark Policy, specifically their list of trademarks. The reason for trademarking is actually quite realistic. The point of a trademark is to communicate who is providing a product or service. By downloading Firefox, you're supposed to know where it came from.
Someone else is welcome to take the code and modify it, but they should release it under a different name to distinguish it. There would be a danger if someone took the Firefox code, put in a vulnerability, and released it as "Firefox" and distributed it, making everyone think it was from Mozilla. Mozilla has a reputation, and over time people start to trust it. It's that reputation that trademarks protect.
Yes, the name of an open-source app can be trademarked. As an example, see Linux itself: http://assignments.uspto.gov/assignments/q?db=tm&rno=1916230
I hope this also answers your second question :)