Problem solved, and I'm embarrassed, to say the least.
I never suspected the router, because I had full internet and Eddie access on one PC, and internet access but with some sites blocked and no Eddie on the other 3 PCs. Also, the problems were similar to those caused by certain viruses and, as Staff pointed out, a faulty TCP/IP stack. However, when I was in the living room unfreezing the Acer laptop, a few feet away from the Vodafone broadband router, I thought "what the hell" and powered off the router for a few minutes. And, you've guessed it, when I switched it on again and the internet connection was restored, the Eddie problem had disappeared. I've tested all 4 PCs, and I can login to Eddie on all of them and access the internet.
Thanks for all your help, Staff. I'm sorry it was wasted effort, but perhaps this topic will help another member in the future!