I am moving from Los Angeles to Moscow. I need to find companies that
offer the easiest, most reliable, quickest way of door to door moving.
What would be the smallest cargo shipment they offer? Do they deal
with the customs themselves or do i have to do it from here? Do i have
to pay anything on customs for my furniture or other items? How long
will it take?

You may find some of the links here to be helpful:

Employee Relocation Providers
http://answers.google.com/answers/threadview?id=189198

Check this site 

www.meest.net,

I used their servece when moving to Ukraine and it was fine...