Ok, it looks like we have overcomplicated this, so I apologize for this.
The answer for your question: no. Everything is on your server. We are downloading the files from a remote server when we install a theme, but from that point on, the files are completely on our server. However, as for plugins, themes also get updates occasionally, and when that happens, WordPress notifies us, and the updates can be downloaded and installed automatically by WordPress, we just have to click a button. And this is a good thing.
Whenever you update your theme, WordPress deletes all the files in your theme's folder, and downloads the new version's files from the central WordPress server. And there lies our problem, because WordPress has absolutely no way of distinguishing our modifications to the theme from the original files, and it deletes everything on update. This is why people use child themes, so you update the parent theme, but everything you have modified, stays intact. Now here the problem is, that the designer who created the Church theme, has also done this, he also created a child theme for a theme called omega. And he named his child theme Church. So basically when you have downloaded the theme, you have downloaded it together with the theme omega. So Church is a child theme, and you cannot create a child theme for a child theme.
So I am not 100% sure about this, but because the Church theme is in the WordPress central theme database (meaning you can find it if you search for it in WordPress Dashboard->Appearance->Themes->Add New Theme), it will get updates eventually from the designer. And when you update, you would loose any modifications you have made to it.
Considering this, I only see one option. Because the Church theme is already a child theme, you don't have to create an other child theme, use it like THAT is the child theme, add modifications, and so on. Of course, the style.css file is not empty, so just add your code to the end of the file.
The most important thing is to never update the Church theme, if you get a notification that it has updates. This way, you can make sure that the modifications you make to the Church theme won't get overwritten.
I hope this makes sense, if you have any questions, feel free to ask.
Toth Balint BT