While applying Scrum, the product backlog items are users stories. I have created a sample story on TFS as follows:
As a user, I can signup to system.
I then created the following tasks:
Some of the tasks are frontend (HTML, CSS, etc) and some are backend (send email, etc).
No. Agile focuses on delivering working code to the customer. Without each part implemented the code written offers no value. Unless the tasks can be shown to offer independent value, then they should be kept together.
Yes the different parts can be implemented by different people. You need to make sure they coordinate appropriately so that the requested feature as works a whole.