From my experience, it certainly isn't impossible to close close quickly and while still overseas. It just really all depends on the landlord.
All documents and applications can be signed either electronically or by hand (printed, signed by hand, scanned and emailed or faxed) and will be sent back to the realtor you decide to use. The realtor would be either the listing agent of the property or one who solely represents you (a buyer's agent).
What a landlord will require (typically) is a rental application that gives general information about you, the tenant, and a credit/background check. If your credit and background check come back to the landlord's liking, they will usually rent it to you as soon as you can move in (they do not want their rentals sitting unoccupied; so the sooner, the better for landlords).
What you will need to pay is the fee for the credit and background check, a security deposit (which cannot exceed two month's rent, and is typically equal to one month's rent), and either a proration of the rent due for the remainder of the first month you will be renting or the first month's rent (sometimes also the last month's rent).
All-in-all, rentals can go as fast as a week or two, which is preferable for landlords.
I have a close friend in this industry and she will tell you it's big here. I would check back when you are closer to moving to see how the area is doing. You can also look at companies online now to get a feel for types of jobs out there to see if there are similar jobs to what you are currently doing. This is what I did when I was moving back here from Maryland years ago.