No, in general people don't. You're right, you type using hiragana, and the software on your device (pc or mobile) will bring up suggested kanji for what you typed in hiragana.

While this keyboard looks really cool, and I assume it's just for aesthetics, the kanji used for the keys doesn't make much sense (ex. Hand, Moon, Bow, Rice Field, Person, Heart, Bamboo, Woman, Divination/Katakana for 'TO').

Generally Japanese keyboards will only have hiragana (た て い す な に etc, like EN has qwerty) and you can swap between hiragana and katakana like you would swap to uppercase with caps lock. Hope that makes sense.

Good luck with your studying btw!