Please Check my Tire Logic....

My questions:

1) Do you think I am good to go with this load range/size? Seems to me they should work just fine given the weights I am working with.

Always go to the tire mfg web pages for actual tech info.
The R250 is a top choice for any trailer.
The LT225/75-16 E has a 2680 lbs load capacity for single tire position. Four of them will give you 10720 lbs of capacity for a nice reserve capacity.
The 2470 lb number is for a dual position or two tires such as a DRW truck and doesn't apply to your tandam axle trailer with a single tire at the end of each axle. Number's I gave are straight from Bridgestone web pages and really are no different than other LT225/75-16 E tires regardless of tire material or brand.

Wally World now has them for $214 a piece with free shipping to store (pretty great price). Will not be using the trailer for the next 5 months. Should I wait till spring or go for the good price that Wally World has on them? Will being installed and sitting for 5 months affect life of tires?

I use LT tires and my 11200 lb RV trailer sits for a good 5 months every winter. I generally get 6-7 years and 50k-55k from them so sitting 5 months IMO not going to hurt the tire.
I can get the same tire from Hibdon Tires Plus stores in my area for 214 bucks but mounting/balancing/stems/etc is another 18 bucks per tire.
Now the bigger question with the prices of crude going up so will the cost of tires.
Your call there.