Timely Comics isn't in the public domain. Their characters and stories have ongoing copyright protection. This means you can't use them freely without permission.
Timely Comics was moderately popular. It had a decent collection and attracted a fair number of customers, but it wasn't the most renowned in the area.
I'm not sure specifically what 'Timely Errors' refers to in a Harry Potter fanfic without more context. It could be the name of a unique plot device, like a spell that goes wrong at just the right or wrong time. Maybe it's about mistakes characters make that end up having significant consequences in a timely manner within the story.
Well, staying relevant to current social issues or trends can make a novel timely. Also, addressing contemporary problems or capturing the spirit of the times in its themes.