Network Theme - Header problems

I am having problems configuring the header of the Network theme. There are four interrelated issues.
1. If I leave the Site Title blank, the word "Network" shows up in the header. Is there any way other than putting a space in the Site Title to make this not happen?
2. I have tried to put an image in the Site Title using html(<img src="url" alt="some_text">. The image appears in the Theme/Customize preview, but not in the live site. Any ideas?
3. Without the image in the Site Title, the Join button appears in the upper right, but creates a lot of white space above the normal header image. Is there any way to avoid that white space?
4. When I try to put an image in the Site Title, the Join button on the right side of the header disappears. Assuming there is some way to get the image in the Site Title, how do I get the Join button to display?