My understanding is that if the phone supports iOS 12, then Apple will not allow you to install iOS 10. If you already had iOS 10, I think it would allow you to do updates within iOS 10.
The 5s does work with iOS 12, so I don't think you can go to 10.
https://everymac.com/systems/apple/iphon...specs.html
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iPhone 5s
Pre-Installed OS: iOS 7.0 (11A466) Maximum OS: Current*
Details: This model is fully supported by the last version of iOS 9 as well as the last version of iOS 10 with the exception of the minor Raise to Wake feature. It is supported by iOS 11, but it does not support person-to-person payments or Apple Pay Cash nor does it support the minor "Appointment confirmation" feature. It is supported by the current version of iOS 12, as well, with the exception of the Memoji, Camera Effects, ARKit 2, and Siri Suggestions features."
As to your question about the installer, I believe that Apple's iOS installer is smart enough to figure out whether it needs a full install, or just a version update. So, no, I don't think you would need iOS 10.0.0 first. So if the iOS 10.2.1 link works, it should give you the full installer. Make a backup first, then try it. But don't be surprised if it does not work.
No expert on this, but I think the only way around Apple's forced upgrades is to have a jailbroken phone.
Good luck.
- Winston