First of all, you're asking a legal question, so you really should be asking a lawyer.
But in answer to your questions:
1a) The issue here isn't copyright. Copyright applies to text. The source code of a game is copyrighted, but I assume you didn't copy that.
1b) Yes, simple games can be protected-- but the nature of this protection would be in the form of trademarks or patents. I assume that the "Cease and Desist" letter they sent you will have the details of what kind of a claim they are making.
2) Depending on how a patent or trademark is written, it certainly could apply to online versions as well as booth versions.
3) How large or small the modifications would need to be will also depend on the specifics.
In other words, without more details, there's not much we (or a lawyer) can do to help you.
However: I'd be willing to bet dollars to doughnuts that your problem stems from the fact that you use the name of their game ("Stacker") in the description of yours. That's how they found it, and they're going to argue that it would cause confusion in the minds of the consumers.
If it were me in your shoes, the question I'd be asking myself is: is the revenue being generated from this game worth the hassle of talking to a lawyer? If not, I'd take down the game, and move on. If so, I'd call a lawyer.