Internet: The Trinket cannot talk to the Internet directly, there are not enough pins and program memory. You can use the Trinket to communicate with another microcontroller or computer to provide Internet communications. For example, a Processing sketch on a computer running Windows/MacOS/Linux could read the alarm state via Bluetooth serial and communicate out to the Internet. There are also many programs demonstrating an Arduino Uno with Ethernet Shield talking to the Internet. A simple Uno sketch to get the Trinket alarm state via a Bluetooth (say an Adafruit Bluefruit EZ-Link shield) receiver and send it to a web page, or service such as xively.com or IFTTT.com via an Ethernet Shield.
Noise: A Piezo buzzer will alert you to an alarm. A loudspeaker would jar you out of bed. Note, if you do use sound, you will want some type of cutoff switch to mute the sound when you do not want it. Or you can modify the code to read from a serial connection and toggle a variable which allows / disallows the sound to be made. If you use a high current horn, connect to the Trinket pin via a transistor to switch a relay to control the horn line. If you connect high voltage or current directly to the Trinket pin it will destroy the Trinket.
If you prefer not to use a Bluetooth connection, you can consider a hardwired connection to the monitor room. This could be in one of several forms - TTL serial direct from Pin 0 as configured. Or you could have a piezo speaker in the home to alert you like a smoke detector. Even a flashing light (preferable for those hearing impaired) via a PowerTail switch. Once you understand the basics, modifying to suit your own needs is relatively straightforward.