There is nothing stopping you from putting in any manufactures HD (or SSD) as long as its SATA II (3.0 Gb/s).
You are noting an issue in a different series of iMac's that use an internal thermal sensor within the HD. Apple has used three different methods within their iMac's to access the HD's temperature.
This model uses a thermal sensor that is stuck onto the hard drives case. Review this IFIXIT guide: iMac Intel 20" EMC 2266 Hard Drive Replacement. Jump down to Steps 19 - 20 where you'll see its removal from your current drive.
As to putting in a new drive:
You'll need to get a HD (or SSD) that is able to run in a SATA II (3.0 Gb/s) system. The Seagate SSHD IFIXIT sells auto sets to the systems SATA port speed. Not all drives are able to run at SATA II so be careful!
But before you do anything make sure your systems firmware is up to date. Follow this Apple T/N: About EFI and SMC firmware updates for Intel-based Mac computers.