1) Extravaganza = an entertainment show of lavish proportions, with lots of characters, lots of costumes, many acts.
Touring the world = a travelling show.
A quarter of a century = 25 years.
I joined the cast = this person became a part of the travelling show.
2) Orient = the East (Asia). It's the opposite of the Occident, or the West (Europe/North & South America).
3) Series of one-night stands = many appearances in different locations, lasting only one night at a time.