How do I remove the “pick 100” box?

The feature was added by the App. The “Pick 100” option is another option to help you solve for a word. When you select the “Pick 100” option, it will allow you to choose a letter position of your choice, to reveal the letter. It is similar to the “Hint” option. Remember, using the “Hint” option randomly selects the letter position/letter. To use “Pick 100” option, 100 coins will be deducted from your Coin Total.

I don't want it. It is too close to the circle. I don't want to accidentally hit it. How do I get rid of it?

You can’t.

It’s just their way of cheating you out of your points when you accidentally press the faux feature

"Hint" isn't random. It reveals the first letters of any unsolved words. Once the first letters are revealed, it then starts to reveal the second letters etc.

It’s a pointless feature. Do they want less people playing word cookies?