Android - How can I be informed through my computer if a new SMS is received?

If you have Android, try Texdro. I tried a few of the other related apps and this worked the best for me. It allows you to send and receive texts from your computer. It also allows you to silence SMS notifications on your phone while the program is running so you can just be notified on your computer (or you can be notified on both). The free version of the app allows you only to connect via WiFi and the Pro version adds USB and Bluetooth options. I will probably be purchasing the pro version when I get some handy $$, mostly to support the dev, because WiFi connectivity works fine for me.

Texdro - Desktop SMS for Android on AppBrain

Android Market QR Code

There is this similar question in SuperUser: https://superuser.com/questions/86076/send-sms-from-pc-through-android-phone

gtalksms allows you to send various commands (including sending sms) by sending a GTalk message to yourself which will trigger various events; note that you need a 3rd party Gtalk software (e.g. Pidgin) since the official Google Talk client does not allow you to send message to yourself, or alternatively you can use a different account for your phone. Download link (open in Android) or scan QR:

also, if you're using Android 2.2 or higher and have the ChromeToPhone apps (open in Android) installed, you can send arbitrary text in your clipboard (Copy-Paste buffer) to your phone using FoxToPhone (open in Firefox) or ChromeToPhone (open in Chrome) extension. Using FoxToPhone, right click on the Send to Phone button and click "Send clipboard", and whatever text in your clipboard will be send to the phone's clipboard, which you can paste to any textbox in Android by Long Press > Paste. Download app (open in Android) or scan QR:

Sounds like Android Notifier delivers part of your request -- notifications and text of messages on your PC. I don't see the capability to reply, unfortunately. That seems a good feature though, perhaps you could request it of the developers.