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The link that you providing is to single gateway, make sure to use the cluster procedure. (its the same document just little bit more below)
R80.40 GA with latest GA hotfix is the recommended version.

Regarding upgrading mgmt server, check that you have no locks in sessions etc.
Publish or discard them.  Also verify that everything is working correctly before you start.
Meaning that you do get logs, able to push policy etc.
Also write down any special configuration that you have made in files, such as user.def file or similar 🙂

If you would go for R81 mgmt server, i would use the inbuilt new upgrade tool within R81 from the dashboard for gateway upgrade.
Its built on CDT and seams to work really good. have tried a few times now 🙂
Not sure if i would go to R81 for gw just yet.