Your statements in post 3 are wrong, and a common misconception. "Shutter speed controls ambient, aperture controls flash."
No, no, no, no, no!
Ambient exposure is affected by the entire exposure triangle. Flash exposure ignores shutter speed but is still affected by aperture and ISO. If you adjust aperture because your flash is too dim, you've also adjusted your ambient exposure.
People fall into "aperture adjusts flash" thinking because with a flash on manual power, you have to set a specific aperture for correct exposure. Still, you can use a different aperture by using a different flash power.
If you want to make the background disappear into black, move your subject away from the wall and place the light rather close to the subject. The light falls of by the inverse square rule, so twice the distance has 1/4 the light, 4 times the distance has 1/16 the light. Proper exposure for a closely-placed light will have very little light for the background. On the other hand, the falloff is apparent within the area you intended to light, as well, i.e. one side of the face darker than the other.
Close light should have some kind of modifier, such as softbox or reflector, to soften shadows. (Well,
any light should, but close light absolutely needs this.)