You can start multiple instances of Zotero at the same time as long as they're pointing to different profile and data directories. On Windows and Linux, the -no-remote command-line flag may also be required.

By default, when running multiple copies of Zotero (either by using multiple profiles or running Zotero in separate OS accounts), the Zotero Connector will save items to the first opened instance. To point a specific Zotero Connector installation to a specific Zotero profile, you can set the extensions.zotero.httpServer.port hidden pref in Zotero and the connector.url pref in the connector. Incrementing the port number by 1 for one pair (Zotero profile + connector installation) is sufficient.

At this time, it's not possible to point the word processor plugins to a specific instance of Zotero.