Are there five consecutive primes in arithmetic progression?

Yes for the lengths you ask for this is known to exist.

The minimal example is $9843019+ 30 n$ for $n=0,1,2,3,4$ (taken from the page at the end).

A more common way to phrase this would be to ask about (five) consecutive primes in arithmetic progression.

Indeed, it is conjectured that there are arbitrarily long (finite) arithmetic progressions of consecutive primes, however this is open. (Without the restriction of the primes being consecutive primes this is known by a well-known result of Green and Tao.)

The longest known arithmetic progression of consecutive primes has length ten.

For further details one could start at http://en.wikipedia.org/wiki/Primes_in_arithmetic_progression (see the section towards the end on consecutive primes in AP) for record data related to this see http://users.cybercity.dk/~dsl522332/math/cpap.htm

Yes. In fact, there is a set of 10 consecutive primes that are in arithmetic progression. I believe this is the longest known set. See here.