When I click to open some applications, they simply start opening and then stop or throw me out of a browsing session abruptly

Update or reinstall them.

Try a shutdown and reboot if that dose not work Check for and install any updates. The third step would be to backup the data and reinitialize the iPad adding your applications back one at a time to see if it is a single app causing the problem. If you have problems with just IOS installed get it in to your nearest Apple store or authorized service center it.