Does the Apple iPhone 5S use the same charger as an iPhone 5C?

Also, if I backup my iPhone 5C, will I be able to transfer all my text messages, contacts, photos, and notes to the new iPhone 5S?

Starting with the iPhone 5 which as released in 2012, all iPhones use the the Lightning connector. There's no such thing as an "iPhone 5S charger". It is simply a Lightning cable.

And yes, you should be able to transfer all of that to your 'newer' phone. By default, all of that is already up in the cloud through iCloud. You would simply need to login with your iCloud account on your 'new' phone and it should automatically re-download all of your data. Though, you may want to make a backup on iTunes on your PC first just to be safe.

Possibly so or not

Yes
Yes

Hi yes exactly the same one.