Do you stone people for not observing the sabbath?
No, but as I said above, it's not because of any moral problem against it. The law that says that people who break the Sabbath are to be stoned is still 100% in effect. The reason it's not used is because there is no Sanhedrin, so no one in this time has the power to impose a death penalty. Once the Sanhedrin is re-established, the death penalty will come back into practice.
And if Judaism doesn't change, then how do you explain the various sects that branch off?
Charedi, Hasidic, Sephardi, Ashkenazi, Lubavitcher, Orthodox, Modern Orthodox etc.
If there is nothing different about these sects then WHY do they exist?
Not to mention that the Yemeni Jewish practices have diverged greatly from that which you know. Of course, it's often more convenient for the orthodox to simply pretend they don't exist...
You named seven 'sects' up there, which in reality are only 2. It's quite possible (indeed I know many people who fit this description) for one person to be Charedi, Hasidic, Ashkenazi, Lubavitch and Orthodox. The only ones of those 'sects' which are mutually exclusive are Modern Orthodox - Charedi and Ashkenazi-Sephardi.
The difference between modern orthodox and charedi is simply a matter of stringency. The laws are the same, it's just charedim are far more stringent in their application of the laws.
The difference between Sephardim and Ashkenazim is geographical. Again, the laws are exactly the same. There are very slight differences in prayer texts, a number of differences in non-mandatory customs and some differences over stringencies (Sephardim tend to be more stringent) but in reality there is no actual difference in belief or application.
I could go into much more detail about all of those sects you mentioned, and demonstrate exactly how someone can be a member of 5 of them, but it would get us very off-topic so I won't.
Yemenites are a different story altogether. They were cut off from the rest of the Jewish world for so long that many of their practices did change, that's true. (Amazingly though, many stayed the same despite the vast time they were secluded.) Now that they are being re-integrated into the general Jewish community, things are changing back. Although it is far more complicated than that, again, off-topic.