What is the most popular Processional piece?
In my experience, The Prince of Denmark's March by Jeremiah Clarke (also known as Trumpet Voluntary by Purcell) is the the most popular by far.  It was performed at the wedding of Prince Charles and Lady Diana Spencer on July 29, 1981.

What is the most popular Recessional?
The Hornpipe from the Water Music by Handel.

Do you use a contract?  Is a deposit required?
Yes, a contract is always used and although a deposit is not necessary, full payment is required before the ceremony begins.