when we went from apex 4 to apex 5, we had to access all our apex 4 apps in a developer instance (which apparently did some conversion/upgrade), and then export them and re-install them in our runtime-only apex 5 production instance. is this the case with an upgrade form apex 5.0 to 5.1.2? or will our apex 5.0 apps just run on a 5.1.2 instance?
Yes, that's the plan.
All applications should work as they did in previous versions. This is made possible via a number of features, one being compatibility mode in app settings.
See here for more detail about how APEX does a pretty good job at leaving existing apps alone. http://www.grassroots-oracle.com/2016/12/phased-apex-migrations.html
This doesn't mean to say you can skip regression testing or expect everything to run smooth. It can depend on how you've built your apps.