This is a very old version of Discourse so it’s good that you’re looking to update!
To start updating from a version that old you’ll want to take a backup of your current site as-is, and store that somewhere safe. Then you’ll want to create a new server with a fresh Discourse install. Once the new Discourse install is running, you’ll want to restore your backup there and verify everything looks ok. Once it’s in a good state, you can switch your DNS to point to the new server and decommission the old one.
In the time since that version, the Zendesk integration has been included with Discourse installs by default, so you’re no longer getting updates in your current state.
Updating is also important because Zendesk is removing the ability to authenticate with API tokens within the next year, Managing API token access to the Zendesk API – Zendesk help
No, we no longer support the version you’re on and it’s not receiving updates.
Yes, we have some documentation here: Enabling Zendesk two-way synchronization
You’re not on our hosting (we update sites on our hosting automatically) so there’s no plan or payment required to update when it’s hosted on your own server.
If you wanted to switch to our hosting you can reach out to team@discourse.org and we could walk you through how to migrate you data, you’d require the Business plan to use the Zendesk plugin.
I believe all Zendesk plans have API access available, but rate limits vary so the required plan depends on your usage.