For architecting, I don't use paper because it is not maintainable, but I use LucidChart and C4 models which I found more convenient. I wrote this post some days ago about that. It's easier to create, communicate, collaborate and maintain. For business processes, I use Camunda BPMN modeler, which is pretty simple yet useful and free. For wireframes, I use Pencil Project.

If your API is pretty simple and doesn't require an ORM, complex routing, and some other things, them Slim is the way to go. Otherwise some more bloated frameworks like Lumen would be okay.

For me, referring the request number is enough. If I want more detail, I take a look at the request itself.