For your first image ...
+1 Christopher's comment ... Eevee - Alpha other than Opaque under material properties.
For your second image ...
It is correct solution of mixing if logo will the same material (like fabric diffuse kind)
For your fourth image ...
Your Mix Shader is mixing Principled and Transparent shader. With Factor set 1 means upper socket with Principled is blender's logic bellow Transparent plugged into lower socket that is handled by blender as layer above. Since Factor is 1 it fully propagates shader plugged in lower socket.
Image texture node (or its modified string) should be connected to Factor socket of Mix Shader node used as Factor (mask).
Specific to your needs - I don't thing you need Transparent shader at all.
It depends on your needs or chosen workflow, but you would be just fine with
- setup in your second image or
- two principled shaders (one for fabric material and one for logo material) and image )logo) use as factor for mixing these two materials.
To animate appearance of several logos I would create black and white animated texture (video or image sequence) and use it as factor for material mix.
Or you can do it in blender by keyframed Factor value of Mix > Colour node ...
... and use it as Factor (mask) in the same way. Also here you can animate transitions drive by some texture node or any other factor.
Or you can render object without logo (fully fabric, no transparency), than render object only with logo with transparent rest of the scene (each logo = solo render animation) and mix them in post pro (compositor) with specific transition.
Range of possibilities is too wide ... :)