I have about 6 of these devices sitting in a cupboard. Not sure what to do with them other than collect them. Back in "the day" I used iPaq's / PocketPCs quite a lot. Some models had SIM card slots and doubled as phones (although the battery life was terrible with radios switched on). But to answer a few of your questions...
So my question is, can I do anything useful or add additional software to the device?
You can certainly install software. With a bit of Googling you might be able to find some old Abandonware sites which cater for Windows CE devices.
Will Linux or Windows connect to the file system through that USB cable or do I need an older computer with Serial port?
I don't think you could access the on-board memory directly via a drive letter like you can with modern smart phones (excluding Apple). You could either copy data to the device via a CF/SD memory card or sync to a PC using
Microsoft ActiveSync which was freely available (and still is, although it's not updated or used anymore except on legacy devices). The split DB9/USB cable does the same thing, it just offers you a choice depending on what I/O ports your machine had back then. ActiveSync supported both. Apart from increased transfer speed on USB, it didn't offer you anything extra.
Even if I did install Linux, what would it offer me? There does not seem to be anything other than Bluetooth connectivity and even then I am not sure for what purpose. Other than Syncing to some desktop windows software to let you work on the go, it does not appear to have any way to update anything until you dock it again (at which time I presume it syncs and sends all queued messages and updates the schedule and so on).
Installing Linux would probably achieve very little, in fact most popular flavours probably won't run on it at all since those earlier units used a 'Digital' StrongARM processor. You are right, the iPaq's were designed to "take work with you". There was limited (or no) online connectivity while out and away from your desktop PC. Back in those days even if you did have a SIM card installed or have your device paired to your mobile phone, mobile data was expensive and slow. You might use it to send/receive e-mail and maybe check the occasional website but that's about it. Bluetooth was primarily used for data transfer between other mobile devices or laptop computers, it was basically a replacement to
IrDA. You could however share your internet connection with your desktop PC at home while it was docked.
All things considered, there really isn't anything useful you could do with it apart from maybe a cool looking retro desk toy/calendar/clock (take it to work and see how many people ask what the hell it is). It's just a bit of old gear with nostalgic value for some.