Yes you can put a new HD in your system without having your old HD. You may need to buy a new copy of OS-X if you don't have the grey recovery CD/DVD that came with the system.
If you don't you'll need to order them from Apple. One more option here is after you install your HD go to an Apple Store and see if they are willing to help you re-install the OS from their servers (you'll need to make an appointment with them and they may charge you a small fee). But you need to get the system functional (boots up with a question mark)
I noted you have listed two HD's on your list is that a typo? You only need one. Depending on the model of HD you bought you may need to alter the SATA I/O speed most of the newer HD's support the newer SATA-III spec (6Gb/s) while your system only has a SATA-II (3Gb/s) interface. Some have a jumper to lower the speed down to SATA-I spec (1.5Gb/s) you may need to do that here. Make sure you check the EFI firmware as well review this Apple TN EFI & SMC firmware updates
Follow the iFixIt guide on replacing your HD iFixIt HD replacement guide on how to do it.
As to the applications you might have had on your old drive. After you get your new HD and the OS installed. We can review what you could try doing to recover some of these without putting your system at risk. Also we can go over how to protect your system. But for now lets focus on the HD replacement.