A number of people have tried to analyze the language you're describing (it's called "lolspeak), but I don't think there's any consensus on the exact rules for writing it. The suggestions you make are all good (dropping "s," dropping the verb "to be," etc.), but there are also a lot of examples of lolspeak that don't use these rules. For example, the original and most famous lolspeak phrase is "I can has cheezburger?" This example doesn't drop the verb "to be," although it does use the wrong form of the verb. I think the only really solid "rule" is that the grammar should be non-standard, but still understandable. Your example lolspeak sentence works very well. :)

By the way, the word "speakable" means "able to be spoken." It doesn't mean "able to speak." For example, if a word is not "speakable," that means nobody can say the word (it's impossible to say it, or no one is allowed to say it). If a crime is "unspeakable," it is so bad that people don't even want to talk about it or describe it. But a person or animal can't be "speakable" or "unspeakable." I think the adjective you're looking for is "speaking" or "talking," which both mean "able to speak/talk." The more common word is "talking." For example, "I have a talking horse" means "I have a horse that can talk." So it would be better to write, "I see many talking cats and dogs on the internet use bad grammar."