6 blown tires and bent axel- help!

op says wrote:
1. Could the axles have been poorly aligned at the start? (manufacturing defect?)

Thats always a possibility but odd tire wear patterns will show up quickly if the axle is bent.

2. Our dealer now tells me that the right side is noticeably lower than the left and that the axle(s?) is/are bent. He suggested that when the tire blew the trailer dropped and caused the bend. Does this sound possible?

The dealer simply doesn't know what he is talking about if he claims a blown tire can cause a bent axle. The trailer doesn't suddenly drop as the flat tire is still carrying a load. Thats how internet myth starts.
One side lower indicates one side springs are weak/one side carrying more load/etc.

3. I bought some parts at a dealer not in our area (fixing a broken water connection that was a result of the blown tire). The repair tech told be that some dealers can test for alignment of the axles. Anyone her of this service/equipment?

Yes some RV dealers are set up to check and align bent trailer axles. Most are not.

4. The cost to replace both axles and get things back in order is going to be about $2000. What would you do- Repair, trade or raise a holy stink?

Find a big rig repair shop that repairs OTR semi trailers. They can check your trailers axles and bend them back into proper alignment. The last trailer I had realigned was 8 years ago on a triaxle GN trailer with 7k axles. Cost was 175 per axle which included new pins and bushings.