These were reported as bug to ESRI:
BUG-000094685 - Keyboard issue in Query widget in Samsung Galaxy 2.
BUG-000094469 - Keyboard issue in Search widget on large Android tablets.
Anyone feeling energetic enough can also try fix the problem themselves. Here is an excerpt of my conversation with ESRI to help folks get started:
"I was able to test this issue in Microsoft Web Matrix until my free trial of Browser Stack ran out. I was able to isolate the issue to the 'jimu-ismobile' class in the jimu.css file. If you wish for the keyboard remain open you need to navigate to the ConfigManager.js file and replace the code in the _isRunInMobile function with "return true;". The root cause of the issue is that the Android keyboard appears when an input element comes into focus. However the Android keyboard causes the DOM to be redrawn, removing the focus from the input element. This causes the keyboard to immediately collapse..... If you wish to have a more permanent fix you will need to wait for the bug to be resolved."