1. What should the organization archive today so that an independent verifier can check the proof years later, even if the original calendar server is gone?
You must ots upgrade a day or so after the original commitment to get the information needed to validate it independently.
If you don't do that it might be impossible to validate it later.
2. Once the proof is complete, what trade-offs do you see between verifying it with your own Bitcoin node and using a public verification service?
Depends on the application -- if the correctness hardly matters and you're just checking to make sure that it didn't fail to get timestamped, then do whatever.
If someone is going to testify in court about the validity of a timestamp they better have checked it against their own node in *addition* to other sources or they'll risk being taken apart in cross examination with the possibility that the whole thing was forged. It's easy to do, so there is no excuse not to.