I wrote a script that uses the Google Images JSON API to automatically fetch thumbnails for posts. I'm currently linking directly to the thumbnail (eg. http://t3.gstatic.com/images?q=tbn:ANd9GcTok4m3DWNRv8gxMDTE0bwj8m-jYl2UGdlbc7ig158m0XosD-lcQEIFcg). Does Google allow that?
If not, I should be allowed to download the thumbnails to my server right?
Its all about traffic. If your app will make tons of traffic, they can ban your server. Anyway, better/best way is to ask them about this subject.
Also this might help you : Google Terms of Service Highlights