I can't speak to vegetarian restaurants. Wouldn't go in one. As for hiking in Phoenix, most visitors that try hiking in Phoenix try the urban mountain preserves and end up on the nightly news getting carried down the mountain on a stretcher.  Do not overestimate your physical fitness or underestimate the heat factor. It may be late Sept but it ain't cool yet here.  Oh, and don't forget that this time of year the rattlesnake babies are just out of their holes and they strike at anything. They have no rattles yet to warn of danger.

Hiking:
Easy - Juddith Tunnel (South Mountain), Hole-in-the-Rock (Papago Park)
Medium - Hidden Valley (South Mountain', via Mormon Trail), Pinnacle Peak Trail, Lookout Mountain, Tom's Thumb Trail
Hard - Echo Canyon (Camelback Mtn), Piestewa Peak Summit Trail (Phoenix Mountain Preserves)

Vegetarian Restaurants:
True Food - near Biltmore area or Scottsdale (Greenway and Scottsdale Road)
Green - vegan - in Tempe or central Phoenix
Pita Jungle - many valley locations
The Coronado - central Phoenix
Flower Child - Scottsdale or central Phoenix
Fresh Mint - Scottsdale
Café Lalibela - Ethiopian - Tempe

Namaste ( Indian food )  on Bell & Tatum next to Walmart... Great food !!