Can I use a wheelchair on Shabbat in an area with no Eruv? Can someone else do the pushing? What about cane or crutches?

Wheelchair - if you can't walk on your feet, you are allowed to use a wheelchair. With an Eruv - anyone can push you. No Eruv - a Jew is not allowed to push you. For a non Jew - in Carmelit (a rabbinic defined Reshut Harabim i.e. most public domains in the world) he can push you anywhere. In a Torah defined Reshut Harabim he can push you only if you are capable of pushing yourself and you are helping him to push. Cane and crutches - if you can walk without them they can't be carried without an Eruv, if you can't walk without them - you can take them with you outside the Eruv. (Shmirat Shabbat Kehilchata, 18,13; 34,27)