No. It is not a biblical commandment to go to shul, but it is a biblical commandment to keep shabbat. That means no driving.

No, it is more important to observe the Shabbat than it is to participate in the synagogue service. Observing the Shabbat means keeping to the restrictions against "work", which is a Biblical requirement. Judaism defines 39 primary categories of "work" that are now allowed. All relate to the activities that were done to build the Tabernacle in the desert. Each of the 39 groups of "work" contains a number of sub-categories. So, driving clearly was not around at the time the Israelites were in the desert, but igniting a fire (or your car's engine) and completing a piece of work (like closing an electric circuit) were. Now, when you take a bus or taxi, you are not "igniting" the spark or "completing" the circuit yourself. But, a Jew may not benefot from any of the blacklisted types of "work" on Shabbat- regardless of who performs them. Simply put: I cannot have a gentile "break" the Shabbat to do something for me. All that is the strict letter of the law. However, some people feel that they are not yet observant and in getting more involved in their Judaism, they would rather drive to the synagogue and experience/ learn more about Shabbat than sit at home and do various things which break the Shabbat anyway. I think there is some merit in that theory, but people should always seek advice from a rabbi before adopting such an approach.

These answers have been for Orthodox Judaism. Conservative Jews are allowed to drive cars to service. I'm unsure about the bus. Biking was considered ok.

I think the non-Orthodox Jews are allowed to drive to shul. I tend to agree with @kitten. But may be I am wrong.

Traditionally, No Conservative and Reform Judaism would allow such. But in Orthodox Judaism, it won't be allowed. Even though your not technically driving, you should allow someone to drive you. Because of two things, if the person is a Jew, then that would be equal to helping another Jew break a commandment of Shabbat. If the driver is a non Jew, it would be as he is giving a service to you. Just like in Shabbat Kiddush, it states that "on it(Shabbat) you shall not do any work-you, your son or your daughter, your male or female servant, your cattle or the stranger who is among you- for in the six days the Lord made the heavens, the earth and the sea and all they contain, on the seventh day He rested". The cab driver would be considered "the stranger who is among you". Also, all and all, the bus ride and cab ride would cost money, and purchasing and doing transactions on Shabbat is forbidden. So if your not Orthodox, your free to do so. But Orthodox Judaism and the Torah wouldn't allow this.

Not a good idea. You don't want to offend your god. He might get angry with you and cause the taxi to get into an accident.