If I make a web page that lets a browser grab a document from IPFS using https://github.com/ipfs/js-ipfs-api, will that page serve that document as well? How can I let that browser become part of the swarm of peers that provides that document to other users?
As far as I understand - no.
js-ipfs-api describes what you can do (an interface). The real execution happens on a node that interfaces with the network (js-ipfs or go-ipfs). On the other hand you can browserify js-ipfs and you’ll get a regular node (currently without discovery protocols or bootstrap, so you need to connect to a known node anyway).
If I’m wrong then please correct me.