1. You can turn it off completely on the Frontend search settings -> General panel: https://i.imgur.com/y1iJnMP.png
2. Under the Frontend search settings panels you can find all the texts and change them as you like.
3. Yes, on the Theme Options -> Settings icon & Drop-down you can style most of the aspects of the icon and drop-down as well: https://i.imgur.com/RsQNYPT.png
I hope this helps!Best,
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