Yes very important. Ablution is a requirement, not an option for prayer.
Make sure your body, clothes, etc are all clean before making ablution.
Then pray.
Allah loves all things clean and pure.
Cleaning yourself helps remove bacteria. God knows exactly how many bacteria are on your body, and your prayer is going to be exactly as effective whether you're praying with 1,443,321,569 bacteria or only 33,456,701 bacteria.
So no, God probably doesn't care how physically clean you are. But it's certainly a healthy thing to do.
Yes. What many people don't understand is that Urdu is about cleaning/clearing your *mind* not your body. It's a *ritual* of purification that helps you shift to a more lofty state of mind before prayer. It's not about physical dirt; if it were you wouldn't be able to do it with dirt/sand when water isn't available.