No, as mentioned, Airport Extreme is at far end of another building about 30 metres away, that is the issue. I do have a long cable going under floors, the back garden etc and that was hard enough to put in, I don't want to do a second cable! I have also tried home-plugs (internet through the electrical circuit, that's what they're called here in UK), but the performance in this out-building is relatively quite poor for some reason, although good in the house.
I fully understand NOT wanting to run a 2nd Ethernet cable, and I also understand the PowerLine Ethernet adapters not working well.
But you should be able to use an inexpensive 5-port Ethernet switch. You need at least 3 ports. 1 port for the 30 meter cable. 1 port for the Mac Pro and 1 port for the Macbook Air.
Generally the smallest switch is a 5-port. That gives you 2 spare ports for a network attached printer, an Apple TV, a TV that is network capable, etc...
NOTE: If you find a 3 or 4 port that is much less expensive than a 5 port, OK. But since 5-ports are the most common, they are more likely to have commodity pricing. You should be able to get something for under $30 US (not sure what that would be UK, but since many vendors make Ethernet switches, you should be able to find a vendor that offers something that costs less than a cheap dinner for 2 (maybe not burgers at McDonald's, but not much more either).
I have been thinking about getting the Ethernet switch and had a look on Amazon.co.uk for 2-port switches (I figured that was all I needed) but am a bit confused about "managed" or "unmanaged". I saw some with actual switch-buttons on them, but I don't really want to be doing switching manually, if that's what unmanaged means. Can you throw any light on that? I guess an Ethernet splitter is not going to do the job.
A simple "Unmanaged" switch is all you need.
It does not need to have any buttons. All it needs is a power brick to plug into the Mains (electrical outlet for those outside the UK). And enough RJ45 standard Ethernet ports for the 30 meter cable, the Mac Pro and the Macbook Air.
You DO NOT need PoE (power over Ethernet). You need special devices to use PoE, so do not be confused by that term either.
I've been using them for years. In one case it was a similar situation. I had an Ethernet cable running through the dinning room wall down to the basement, across the basement up through the 1st floor hall closet into the 2nd floor hall closet, through the wall into the back bedroom. I had a 5-port switch in the dinning room where my Wife's office was, and a 5-port switch in the upstairs bedroom where I had my office. Both of them were inexpensive unmanaged 5-port switches.
I'm thinking it will be simplest to go for the switch, as you suggest, or for that matter even a basic wireless router, to give my some mobility with the MB Air in that outside room. Just so long as it works!
A basic WiFi router, that you put into "Bridge" mode, would give you a similar result, plus WiFi support in the part of the house. It would just cost more than a switch, but if you want WiFi service in that area, like for you iPhone, iPad, or other WiFi devices, it might be the better option.
It can be fun figuring out how to put some WiFi routers into bridge mode. Apple routers are easy, and you have already done it. Other vendors often go into bridge mode if you turn of their DHCP server. Some need you to turn off the NAT server. Like I said it is different for each, but some Google searching should turn up instructions.
A WiFi router will have a built-in Ethernet switch, so you can run the 30 meter cable into the WiFi router's WAN port, and use one of the switched ports for your Mac Pro, and either us another port for your Macbook Air, or just connect to the WiFi.
Didn't get anywhere with the experiments, I'm afraid... still just "no IP address" or "self-assigned".
I'm sorry you cannot get Internet Sharing to work on your Mac Pro. It really should work and do exactly what you want.
Then again a dedicated Ethernet switch should do the job and not fight you about it.