Shabbat

Is one allowed to use a stroller within an eruv on Shabbos?  

If a Shabbos elevator is permissible [by some learned rabbis] what is the problem with riding a subway when the passenger does not have to pay to enter and is not making or breaking any electrical circuit?

I was recently at a Chabad Shabbat dinner and someone accidentally backed into a light switch, causing one of the young boys to go out onto the street in Brooklyn in search of a non-Jew to turn on the light since doing so himself would break the Sabbath. Many synagogues similarly employ non-Jewish custodians to turn on lights and prepare the sanctuary on Shabbat. These people are often colloquially referred to as the "Shabbos goy." Isn't this disingenuous? Taken to an extreme example, in the case of a murder for hire, the law imputes criminal liability on the person who did the hiring even though they didn't pull the trigger. If a person is restricted from doing a certain act by Talmudic law, how can he legally get around the prohibition by simply getting a nonobservant person to do the thing he is not allowed to do in his place?

Why do Jews observe Shabbat from Friday evening to Saturday evening and Christians observe the Sabbath on Sunday? If Christians incorporate Genesis into their Bible as first book of the Old Testament, why is there a discrepancy?