How to transfer apps, contacts and settings between two identical phones?

If you're rooted, you can do a nandroid backup of the entire phone (it's like taking a snapshot of the phone's data) and you should be able to push it on the new phone so long as it is the same model.

Additionally, if you're rooted, you can use Titanium backup on the apps on your phone: it'll do both the app and the data associatied with the app.

Lookout should do all, or at least most, of this for you.

(Warning: Market-speak follows)

Your mobile device is your most personal computer. It contains your personal data including contacts, photos, video, e-mail, and text messages. Backup your data and keep it safe and secure.

Whether you accidently delete data, your phone is lost, or even destroyed--your data is safe. You can restore any data with the click of a mouse. Changing devices is just as easy and your data can be transferred in seconds.

Keep the SD card when you return the phone and put it in the new phone you get.

Contacts: You contacts are constantly synced with the Google account associated with the phone so there's no need to back those up.

Media: Pictures and Movies should be stored on the SD card which you will keep.

Apps: You can back them up to the SD card with a 3rd party app or sign up for appbrain which syncs with your phone and keeps a list of all the installed apps. You can then reinstall them right from Appbrain.

Settings: Theres no way out of the box to back up phone and app settings.

Other: There are a number of free apps in the market that backup particular data like you SMS messages, your call log, your apps...

Total Solution: There are a couple apps out there that claim to backup and restore the entire phone. I actaully had the same exact problem you are having back in January with my Droid. I used a free version of MyBackup Pro back them but I'm sorry to say I can't even remember if it worked. Its worth a shot but use caution, it seems to have some mixed reviews.