As many times as they want.
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Yes, but that player can also switch any of their letters out for new letters if they decide to skip.
The game has a timer for each player, the nudge and skip buttons are enabled once he/she exceeds the time allotted. then, you can nudge him/her or skip his/her turn.
I think you have to fail about 5 times.
When both players pass their turn three times in a row, the game ends and the player with the most points is declaired the winner. In reality, this only can happen after all the letters have been drawn, because you always can skip your turn to swap letters. Of course, if both players swap the same letters, there is no point at continuing the game.
You will not loose points directly, but you will forever give up the opportunity to earn points for the turn.