Is it permitted to use an elevator on Shabbat when a non-Jew presses the buttons?
It is forbidden to have a non-Jew perform a melacha for you on Shabbat and therefore one cannot accept his assistance in this case (opening or closing the door, pressing on the right button etc.). But if there are Jews and non-Jews in the elevator at the same time and he presses the button on their behalf as well, than you can use the elevator. In any case it is best to refrain from such practice because after the non-Jews leave the elevator, you may be left alone and end up making him press the button for you. Another problem would be if other Jews see you using the elevator but do not realize that there are non-Jews there as well. They may think that it is permitted to use an elevator even when there are no other non-Jewish passengers.

What are the applications of "aino oleh ul shulchan melachim" (cannot be found on the table of kings) regarding food cooked by a non-Jew.
Food that cannot be found on the table of kings means that food unworthy for a king is not forbidden if cooked by a non-Jew. Today most foods can be found on a king's table or at least are good enough to be eaten by him. Therefore this heter is not relevant (there are other heterim).

May I visit a church designated as a museum not only as a place of prayer?
It is forbidden to enter a church whether it is a place of worship or a tourist attraction.

Quite often there are ovens in rented apartments that are probably treif. Can one cover the food with aluminum foil and still use such an oven?
All utensils that actually touch the food must be tovelled. According to some opinions if it is extremely urgent one can use the utensils in a vacation apartment even though they haven't been tovelled. You must remember though that in places like these apart from not being tovelled, the utensils are probably treif due to milk being mixed with meat or non-kosher foods and cannot be used. When using an oven you must completely wrap the food with two layers of aluminum foil and only then cook it.

Can I go to a football match on Shabbat, having bought the ticket before hand? I only need to show the ticket and don't need to carry any money.
If entrance to the stadium only requires showing your ticket, it is permitted. However you must remember that in most places there is no eiruv and it is forbidden to carry on Shabbat. Also consider whether enjoying a football match on the holy Shabbat is appropriate for a religious Jew.

Is it permissible to drink Coca Cola without a hechsher since the drink's secret formula is identical across the world? Likewise are McDonald's chips which are made using the same products in all the branches worldwide, Kosher?
Even if the formula is the same everywhere, there is still need for supervision of the manufacturing process in all the factories (to make sure that the manufacturing rules are kept according to halacha). Therefore you must not eat any food without a reliable hechsher

Since New Zealand is situated close to the international date line (where on either side of this imaginary line the date is different), what must I do on Shabbat or Yom Kipur? Must I keep two days?
There are many opinions regarding this problem. The halacha is that you must keep the day according to the custom of that place (in your case New Zealand). There is definitely no reason to keep two days.

Can you call from Israel after Shabbat to the U.S. while it is still Shabbat there and leave a message that might be heard?
If Shabbat has gone out in Israel you may call the U.S. and leave a message even though it's still Shabbat there. Obviously, the people listening to the message in the U.S. may do so only after their Shabbat has gone out. If the message will be heard during it's recording (in the U.S.), one should refrain from doing so since it disturbs the Shabbat atmosphere.

What is the law regarding an organized tour where the route has been changed making it necessary to travel on Shabbat with a non Jewish driver?
It is forbidden to instruct a non-Jewish driver to drive on Shabbat due to a change in the planned route. The real question is what happens when the majority of the group is not Jewish and they decide to travel on Shabbat paying no consideration to the Jewish tourists' wishes. Must the Jewish tourists remain where they are, or may they join the group? The general rule is that if the driver is going to drive in any case with or without you, you can join him. The truth is that it's not practical to travel this way for two reasons: * You mustn't carry from domain to domain (from the hotel to the street, from the street to the bus) not even with the help of a non-Jew. * Normally we're talking about a long distance (otherwise you could have walked). By traveling a long distance you are not keeping to the Isur of Tchumin (Shabbat geographical boundaries). Therefore there is no practical way to join a group that is traveling on Shabbat and the Jews must remain where they are and meet the group on Motsai Shabbat.

What is the law regarding an Eruv on Shabbat outside Israel?
In a place that has no Eruv one may not move any item from Reshut Hayachid to Reshut Harabim and vice versa. Also one is prohibited in moving an item more than 4 Amot (about 2 meters) in Reshut Harabim. Reshut Hayachid for this matter can be any closed area surrounded by a wall, fence etc. Reshut Harabim can be described as an open area such as a park, garden or road. The laws of Eruv are of course relevant anywhere not only in Israel.