Intro to HCI: Tips & Tricks for Mobile Web Applications

Accessing Phone Location:Use the HTML5 Geolocation API HTML5 Geolocation API
Using Google MapsUse the Google Maps Javascript API

Accessing Gyroscope/Accelerometer Use the DeviceMotion/DeviceOrientation APIs((tutorial))

Accessing Calendar:There's no way I know of to access the native calendar on the mobile device. However, you can integrate with Google Calendar API and load the data that way.
Accessing Contacts:Similarly, no native way of accessing contacts but you can use Google Contacts API if you need their contact info or use the Facebook APIfor friends
Facebook:If you need to access Facebook data on their phone, use the Facebook Javascript API. Otherwise, there are several third-party Node.js libraries to access the API.

Taking a photo/picking photo: If you have a newer version of the iPhone (iOS6+) or Android (3.0+), use HTML Media Capture
QR Code Reading: Substantially more difficult. Just about do-able if you take a photo using HTML Media Capture (see above) and use a QR code library on the server-side to read the QR code from the picture (e.g. https://npmjs.org/package/jsqrcode). No guarantees on this one though.

Web fonts: Use custom fonts for better typography. Google Web Fonts is a great source of easy-to-add fonts (list of highlights). Also, here's a nice article about good typography.
Subtle Patterns: Ever wondered how to make backgrounds a little more interesting than a plain color? You can use Subtle Patterns to add a light pattern to your designs.
Good icons There are a number of free icon packages out there (just search around). The Noun Project is a good start though.