The Twitter Search API don't give back all tweets available for a hashtag, only the latest, exactly how long back "the latest" is, I do not know. So, there might be a chance that no new tweets has been tweeted with the hashtag you use and therefor do not exists in the Search API anymore. Have you tried with another hashtag?
I have never had any issues to use the shortcode on pages. Do you have any custom filer running on the_content for pages?
From what I know and tested you can't have shortcodes in the default text widgets you have in WordPress. So, I must ask how you get a shortcode in a text box widget?
Yes, there is an upper limit, but what exactly it is I do not know and I really do not recommend you to show "all" images as that will set a huge load on your server to load and re-size all those images.