1) Does the Swan and Dolphin count as a Disney resort for the purposes of taking advantage of the early morning access to the parks?
As Naomeri mentioned, there is special morning access for (any) resort guests currently; however, everyone these days has "early access" in the sense that parks are letting people in up to one hour before official opening time. At Swolphin, you can walk to Epcot would recommend leaving your room 75 mins before opening, to be among the first in the park when the gates open. For HS, you can also walk and be the first in, or take the friendship boat or skyliner (transferring at the skyliner hub at Caribbean Beach) which are more scenic routes but will get you in later than most other guests (though still before park open)
2) We're going to Beaches and Cream for the first time! What are some can't-miss dishes?
The ice cream is the real star of the show. The "kitchen sink" is the iconic dessert, but the "no way jose" peanut butter ice cream sunday has a devoted following. For food, it is a soda shop--just your usual burgers and sandwiches. The cheese and bacon tots are probably the most unique thing to try.
4) Talk to me about Satu'li Canteen! I'm trying to branch out of my usual dining options, and this is one I've never tried before.
It's fun and feels like you are eating food from another planet! My DH likes the sweet potato hash base with beef. I am not gonna lie, I think the bowls are a bit weird (the sweet/savory thing has never been my favorite) so I secretly get the kids food for myself. I like the hot dog-in-a-bagel-roll thing and the cheeseburger pods (which are basically bao buns, if you like dim sum).
6) We have early Droid Depot appointments on our HS day. I'm hoping to go on MF:SR this morning, but I'm not sure what to do with the droids while we're riding: take them with us? Put them in a locker? Run back to the hotel? I certainly don't want to leave them in our stroller, so any advice there would be helpful.
Take them with you. You can set them on the floor of the ride simulator--plenty of room. That's where we put our Savi's lightsabers and nothing bad happened to them. But candidly, I don't think you really need to keep those early droid depot appointments. We've done it on two separate trips (got the R unit one time and the BB unit the next) and both times there was plenty of walk up availability--we went in the afternoon and one time it was no wait at all and the other time no more than 15 mins. Would use your morning to ride attractions before the wait gets long and do droid depot when the ride lines are long in the afternoon.
Edited to add: I didn't answer 3 and 5 because I haven't stayed at the Swan and Dolphin.