At ANACS they gave the nice plastic boxes when I submitted 25 or more coins, then I started getting only cardboard boxes back, I called them about it (back when James Taylor was running it) I know James from way back, he said he'd ship me a couple, and they did, but ever since I only get the cardboard boxes anymore. Can't say on NGC or PCGS, today, PCGS used to return every submission over 5 coins with a blue box to me.
As to shipping, I have always used the TPGs preference in flips, and all of them it seems like large Crown-sized flips, ANACS always gives me a buch of them at shows when I stop by their table to drop off coins. These are 2.5" x 2.5" sized and I use a large black sharpie marker to number each coin as it matches the line on the submission sheet (One coin per line, sometimes I've had to skip lines for notes) This allows them to double check faster when my coins arrive. This is how ANACS prefers the coins to come in, NGC and PCGS too. I rubber band the flips into packets, usually 10 per bundle. 90% of the time I submit at a coin show to save the postage sending them in the mail, and the worry of loosing them in shipping.
When mailing (if you do) I would suggest you use only Registered Mail in a good box sealed per USPS guidlines with brown threaded kraft tape. I used to use FedEx, but please note FedEx will not cover coins in the event of a loss (it's in their fine print), even if you have paid for their insurance up to $25,000.00 - you won't be able to win a claim if the box has collectable rare coins or money in it, period.
Back when I was a full time dealer, my insurance rider with my coin insurance company "Hugh Wood" covered transporting and shipping issues. I never had a loss submitting or getting them back via FedEx, or USPS mail - knock on wood.
http://www.hughwood.com/sub_menu.php?submenuId=54Hugh Wood Insurance works with
ANA member dealers and collectors.
Here is NGC's shipping instructions for you, about the same as most TPGs really.
http://www.ngccoin.com/news/viewart...rticleID=561FedEx
http://www.fedex.com/us/service-gui...d/index.html Look at the section on Declared Value and Limits of Liability (Not Insurance Coverage) Part F is the kicker $1000 maximum claim per package. NGC does say they reccomend FedEx, they may utillize their own shipping insurance but not sure how if it's lost by FedEx when you are shipping to them, I know it works when they are returning your order to you. Might want to call them on that one.
Hope this helps you out a bit.

Wanted to edit and add why to only use USPS Registered and no other service they offer
http://pe.usps.com/text/dmm300/609.htmSection 4.1 o
All claims not sent via registered mail are limitied to $15.00 on coins or bullion. Priority, First class, insured or not $15.00 is what it say in the USPS fine print.
"Buy the Book Before You Buy the Coin" - Aaron R. Feldman - "And read it" - Me 2013!
ANA Life Member #3288 in good standing since 1981, ANS, Early American Coppers Member (EAC), Colonial Coin Collectors Club member (C4), Conder Token Collector Club member (CTCC), Civil War Token Society (CWTS) member, Liberty Seated Collectors Club (LSCC) & Numismatic Bibliomania Society member (NBS), USMex, Member in good standing, 2¢ variety collector.
See my want page:
http://goccf.com/t/140440