I can't paint the texture. Although I can't draw, I can fill it all at once with fill icon. I have no idea what that means. I do not understand the relationship between the property panel on the right and the paint screen on the left.
While in Texture Paint mode, with the Active Tool tab open (this is the case in your image), look under the Texture Slot Section for Mode. There are two options. Material and Single Image. Your pic shows Material selected currently. Try Single Image.
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