I would say in the order they were made available on the Oracle support site for download. Most of them are cummulative and include the previous fixes. Refer to read me of the patches and you can judge there will not be too many to apply.
As mentioned previously, you should check the readme of the patches. Most of the time it is cumulative and the Service Packs include all the previous hotfixes/fixpack. In your case the Service pack 1 includes fixpack 15.
Always apply latest service pack first and on top of it apply the latest fixpack available for that service pack. That should include all the fixes.
In a Websphere environment (and likely others) you are effectively being asked to do the following:
1. unstalling whatever version you are currently running
2. installing 8.2.1 (this is a full product build)
3. uninstalling 8.2.1 (this is a full product uninstall)
4. installing 18.104.22.168 (this is a full product build and is a version older than SP1)
except for 3 (IIRC) files copied manually from 8.2.1, everything is kept in your P6.ear which you install.
Good luck with whatever you decide to do.
Is there an msi or executable for applying the SP1 updates to the P6 8.2 EPPM Professional client? Furthermore, does an msi or exe for the application of fix patch 22? Does fix 22 include the features included within fix patch 16?