Does Lima Sky have a case? Maybe. Based on rumours (*) I've seen, they do not. Might be I'm missing some vital information, which would reverse my opinion. However this is related to recent news about Apple removing VLC from AppStore (news via TUAW) due same kind of events.
When one party calls Apple and claims license, copyright, trademark or any other legal infringement, then Apple contacts the other party requesting a common resolve - or the other party's application will be removed from AppStore. Pretty powerful threat, since there is only one legal iPhone distribution method and that's via Apple AppStore. Some might call that monopoly, but I'm not a lawyer.
Anyway, Apple is right. This is exactly what Apple should be doing and any other way would be wrong. Wrong, I say! Common resolve is fair and protects the small guys. Apple, as a small guy I really appreciate this!
But it is a system and any system can be misused for mischief (in the negative meaning of the word). Apple does not have resources nor interest to check each and every case manually and mistakes can and will happen. Seems like Lime Sky is counting on that - or as the discussion forum thread at touchArcade kind of says: first scare small guys out of business, try to get momentum.
What can a small guy do, when he's being bullied? Not sure if there's a case or not, no money for lawyers, Apple not listening, no idea what to do? Running away is an excellent advice in real life, but you can't do that in ...whatyoucallit internet.
Make it public. Make noise, lots of noise, ask for advices. Someone somewhere has been in the same situation before and knows what would they do, if they got a second chance. Lot of people will tell you what they would do in the first place. Everyone will disagree, but when there's enough noise, look for two things:
Tired old guys who's seen it all, done it all, for too many times. They are too old for this sh*t. Listen carefully and you might get some brief solid things to consider. Then look for general common opinion. Usually you can't rely on it, but it's the way the world is going right now. You might go with the flow, hope for the best. At least you'll have support and sympathy. Never underestimate that!
One more thing. Check if you can misuse the system. They did it to you, so that's an invitation to return the favour. Maybe you can call Apple and claim the first party is actually infringing your rights with their app. If Apple takes your word, you won't resolve the case and their app will be removed from AppStore, too. Damage to both parties, but most likely they will suffer more.
Power to the small guys, stick together!
(*) Does Lima Sky have a legal case? Rumours say their trademark application USPTO Serial Number 85079944 for "doodle" was rejected, because "Registration of the applied-for mark is refused because of a likelihood of confusion with the mark in U.S. Registration No. 3327023". I have no idea that means, haven't read any of the related papers (which might be available via United States Patent and Trademark Office).
Btw I'm happy to report that I don't have any Lima Sky applications within the 400+ app I've got from AppStore. Nothing to do with this case, just didn't know they exist.