The downpipe/dump pipe is the pipe from the turbo exhaust outlet to the exhaust front pipe (basically the pipe that vents gases from the turbo into the exhaust), a screamer pipe is a pipe from the wastegate outlet that vents into the atmosphere.

Yeh an internal wastegate is built into the turbo's exhaust housing and uses an actuator to open the gate at whatever the actuator's boost level is (a vacuum hose is connected to the throttle body). The gases from the internal wastegate vent out next to the turbo outlet, so goes into the downpipe.

You're right about the external gates. Internals are usually used on smaller turbos and for lower boost applications. Externals gates generally dont creep as much as an internal gate could, and are useful for higher boost applications.

The way it works is basically this. It is mounted on the exhaust manifold, preferably in a spot where it can get gases from all of the cylinders' exhaust ports. It uses an internal spring and is connected by vacuum line to the throttle body, so when the boost exceed the spring rating the gate opens and vents exhaust gases so the turbo cant spool up any higher.

There are various sizes. For example Tial make 38mm, 40mm and 44mm gates from memory. The bigger ones are required for either larger engines or higher boost applications. You can also specify what spring you want (ie what the maximum boost level is).