iCloud does not store messages for you to see/access at icloud.com. It does back up messages so you can restore your iPhone if it ever needs to be reset to factory default. A backup is an image of all your data on the iPhone, and if you delete some messages, then they won't be part of any further backups.
If two people use the same iCloud account, then their devices are kept in sync, just like a single user's multiple devices are kept in sync. It's usually a good idea to have separate accounts for two people, since it's rarely the case that they want each phone to be an exact replica of the other.
Your iphone syncs contacts, events, camera roll, and many other things without the use of a computer.
Read about icloud here... http://www.apple.com/icloud/