there seem to be an awfully large number of 'defaults'...

Actually, looking a little closer at your original posting (which will
teach me to read before I type ;-) ) I don't think you have any
problems, other than your router not being set up for wireless. The 'no
loopback extensions' is just fine and I would expect the results you got
- 'lo' is the internal 'loopback' interface and eth0 is your wired
connection, so neither should have any wireless extensions. wlan0 is
your wireless interface and is showing that you do, indeed, have a
wireless interface but that it is not connected to anything.

Rather more interesting is the result of the lshw, which shows that your
wireless card *is* connected to something - or, at least, the computer
*thinks* it is. If you aren't connected, then lshw should not show an
ip address for the interface, but yours does - and the fact that it's a
10.x.x.x address, rather than a 192.168.x.x address is intriguing. Both
are private IP addresses, but 10.x.x.x is more typical of an internal
business network while 192.168.x.x is more typical of a domestic network.

But as to what it means. Well... Um... Er...

However, as others have said, your router shouldn't care what OS you're
using and your OS doesn't need a driver for it - the router just sends
and receives standard packets of data which have no tags in them to
indicate what has sent or or is receiving them.

So, a couple of questions.

Have you ever had this computer connected to any wireless network?
Is this a brand new router or a second-hand one?