how to make cursor jump automatically to next input cell after evaluating previous one?

You can setup CellEpilog to automatically advance a cell after evaluating the current one. That way, you don't need to press the down arrow after evaluating a cell.

SetOptions[EvaluationNotebook[], 
    CellEpilog :> SelectionMove[EvaluationNotebook[], Next, Cell]]

If your keyboard has a numeric keypad, or a way to emulate one by using a function key, you can use Shift+Enter on the numeric keypad. This keystroke will evaluate a cell if you're in an evaluatable cell, or move to the next evaluatable cell if you're not. It would still be two keystrokes to move then evaluate, but it'd be the same keystroke.