Try unpacking the textures before exporting. The unpack option in the File / Externa Data can save all the packed textures to a textures directory in the same location as the blend file and the .obj file. The mtl file should then be able to point to those textures.
Similarly I’ve found that for the OBJ files it splits .obj and .mtl into 2 separate zips - how to make one file? Just merge?
Check your obj export settings. What zips ? Post a link to an example blend file that does this.
Cycles materials only work with the cycles renderer so for anyone to see what you see you would have to use blender, so why export as another format.
If you want to export materials in a more universal format with you should export from blender renderer. Even then not all material settinsg are universal across all applications so wht you see in blender is not exactly what you’ll see in other renderers.