First up - click the blue apple in the top left corner and select About This Mac (I think that's still included in OS 10.6). Then click More Info, which will open System Profiler. Then click Hardware (right at the top of Contents). From the Hardware Overview provided you should find Machine Model and Serial numbers.
Once you have that info, use Mactracker
http://mactracker.ca/ to get all the data you ever wanted (and some you didn't) about every Mac ever made.
According to Mactracker, there is no 2008 model of Mini, but we can probably assume that yours is a mid 2007 model.
Re network - I can't be certain without further info, but I think we can assume that the Mini will work alone.
Re Apple modem -
Amazon.com: Apple MA034Z/A External V.2 USB Modem: Electronics (you won't like the price!) These modems are no longer made.
What brand of modem do you presently have? Recognition in hardware may not signify that it will work with MacOS (I have a similar situation with digital camera and data cable). However, we may be able to find a driver or emulator for your modem.
Update since your most recent post:
Firewire and the internet are not related. Firewire is a means of connecting via cable external devices like hard drives, optical drives, another computer, etc. The fact that it's 400 Mbps tells me that your Mini is indeed 2007 (the next, in 09, had FW800).
Modem - Download HP Compaq nw8240 Agere Modem Driver 2.1.51.00 from here
http://downloadnew.org/drivers/modem/hp-compaq-nw8240-agere-modem-driver-107773.html Read carefully the info about which OS are supported.
Bluetooth - I'll leave detail of that to others, but suffice to say that it is a means of wireless interconnection between devices like a keyboard, a mouse, also a modem (but yours is USB). This might assist you
http://gizmodo.com/027755/bluetooth-dial+up-modem.