I've had a chance to investigate and ran through roughly the same configuration as you. I'm using the Twenty Eleven theme, installed Custom Post Widget, created a content block and pasted content with many headings. I then dragged the content block widget into one of my sidebars and it appears where it should be. I then changed my homepage to a static page and can see the TOC get printed out on both the main content and the widget area. I wasn't expecting it be printed in the widget at all as it is a different content type and custom content types are disabled from auto insertion by default.
After further investigation, it turns out the main content of the page dictates the content type for all, eg because the homepage's main content is from a static page, it treats everything else on the page as content type 'page' and loses the content block type. This means the widget will auto insert a TOC if you have page selected in your TOC+ settings.
After discovering this, I began testing your other issues but wasn't able to replicate them.
1. I added [no_toc] to the top of the content block and it suppressed the TOC for itself only. The TOC for the main content on the homepage still appeared.
2. I unchecked headings in the advanced options and they were appropriately excluded.
I'm not sure where to go from here. The only thing I can think of is maybe order will matter so check to see if you're calling the_content on the main content prior to the widget stuff?
Another suggestion would be to try and use a text widget for your right hand side content?