She may have a better experience with Safari (built in browser) for her needs. If she also uses her iDevices, Safari can sync bookmarks, passwords and such between them securely.
The external drive seems pretty versatile (FireWire, USB, and apparently eSATA connectors) but the computer shop was unable to get their computers or SWMBO's computer to "see" the drive, and I have had the same results with my Windows computer. Is this because the drive was originally formatted to run under Macintosh OS?
If it's working, it should show up somehow even if it's formatted for a Mac. On your Windows computer with the drive attached, run Disk Management and see if it shows up in the list. Device Manager should also have an entry for it. Or on the Mac, run Disk Utility and System Information, those are the comparable programs to Disk Management and Device Manager.
I have several suitably-sized 3.5" hard drives and an external USB dock for them. What do I need to do to make the iMac run Time Machine on that hardware?
Plug one in, then when the Mac sees it, it will ask if it should be used for Time Machine. It'll take care of formatting it appropriately if needed, and start the first backup.
If you want to back up to multiple disks, open System Preferences, and then Time Machine. Click Add or Remove Disk to add a new one into the rotation.
To run any of these programs on the Mac, press Command - Space to bring up a search window which can find the programs mentioned above. Or click the magnifying glass icon in the upper right to bring up the same search interface.