1. Do any companies sell non-qualified (not for retirement) immediate (payments start right away) annuities to individuals that are in their twenties?
Maybe. However, given your age, the payments would be really (really really) low.
2. Is it possible to buy a contract for $1000 or less?
$1000 up-front price? Or $1000 per month? I doubt you could find anything for $1000 up front; it would cost the insurance company more than that to set it up, it just wouldn't be worth their time.
3. Will they mail the check to my door or do I have to initiate a withdrawl.
I'm sure they would mail you a check every month or even do direct deposit.