I think AirDroid is pretty fed your needs. If your computer and mobile are connected to the same network in most cases, the free version will work perfectly for you. But if you want the most out of the web-based connection, it is good to pay for the premium version.
Major Feature of AirDroid
Remote Control (AirMirror)
Sync Phone’s Notifications
Messages/Contacts/Photos/Videos/Ringtones/Files/Call Logs/Apps Management