What settings do you have set in ModernMix as the store should not be showing up with 1.1.
Also exactly what were you doing prior to the store showing up and is it showing up fullscreen or windowed?
Regarding the icons, if you open C:\ProgramData\Stardock\PY do you see the correct icons named FileManager and winstore (both with lots of extra bits of text)?
I think the problem with the Store popping up has gone away. It may have been related to rapid boot, as it went away after doing a shift+shutdown this morning and I haven't seen it happen again since. To answer your question though, it was appearing seemingly at random -- while running metro apps or just browsing the web.
Also, the issue with the icons has changed. Today the Store icon appears correctly, but the Photos icon still shows the Chrome icon. In the C:\ProgramData\Stardock\PY folder, I find the correct icons on the named .ico files. I also find 4 Chrome icons named ".ico", "_2.ico", "DefaultBrowser_NOPUBLISHERID!Chrome.ico" and "DefaultBrowser_NOPUBLISHERID!Chrome_2.ico".
Is it safe to delete any of them? I suspect the first two are a problem, while the last two seem to fit the pattern of the rest.