How to read changing feature type name without having to manually make modifications to script?

Part of the answer is in your 2nd screenshot, the generic port. You can specify in the Output section to use a Single Output Port, so that the specific feature layer names won't be shown as an output port. But to answer your question...no, it's not possible to dynamically change the port at runtime.

I tried this, but then the data seems to be sent to the <Generic> output without any fields. Is there a way to sent all of the data, with all the columns, through the <Generic> output port?

You can expose attributes on the Generic port for attributes that you need to work with in the workspace.

When you read data through the Generic port, the fields are still present, they just aren't exposed. But even though they aren't exposed, they can still be written on a dynamic writer, which is generally when the generic port would be used.

@antoine.prince​ The Tutorial on dynamic workflows gives a little more background on this puzzle.