Will I be able to install later versions of macOS on my Early 2013 Macbook Pro (10,2) even when Apple drops support for that hardware? Would I be able to install them with the patcher tool?
It will entirely depend why the hardware is unsupported - with Mojave a lot of it relates to whether the GPU is Metal compatible, as I understand it.
Given that they just raised the hardware requirements last year, you probably have a few years (and thus a couple of new versions) before your machine is not supported.
Not sure about the "few years" time. My Mac mini 2011 refuses install Mojave. I tried a patcher tool, but the results were not good (slower work and some background colors look odd)