Sorry, could you be a little clearer about what shortcodes you are using and having a problem with?
Are you using native WordPress shortcodes alone, and/or plugin shortcodes. Sorry to reiterate this again, but it's still not clear to me what you're clicking/inserting or typing via the editor whilst inside the tab shortcode.
How are the slideshows generated for example, is that a plugin or theme feature? (as it's not native to WordPress to my knowledge)
Would this be an accurate example of code you're having a problem with?
[tab name="Tab"]Native WordPress Gallery here(usually represented by an image in WordPress, which i can't display in this forum)[/tab]
[tab name="Tab"][gallery ids="531,204"][/tab]
I can place a gallery inside a tab just fine my end, so i'm wondering what specific shortcodes or content(with a specific plugin?) i'd need to put inside the tabs to replicate the issue you're describing.
An example like i've given above(trimmed to just the essential/problematic parts) would be most helpful in testing.