Answering my own question... after a day or so, the apple tree grew some leaves. Eventually it grew more leaves, and then eventually started producing apples. It took several days (I didn't keep track). Once the tree starts to produce, it works like a farm (5 people x 1 minute = 5 apples) and appears to be good in all seasons.

No, Razor Shell is a crafted item, made in the Fishing Hut. (It takes a Clam to make it, which you do find by clearing dunes.)

I bought an apple tree, but it isn't giving me any apples. Do they only produce in a particular season?