First, just because the other Internet connection is named Connectify does not mean that it necessarily has anything to do with the software you are talking about.
What type of Wifi router are you using, and more importantly what type of encryption is it using? If it is using WEP encryption, that is old technology and very vulnerable to being broken into. If your router supports it, use WPA2. If not, get a router that does.
Questions about the legality of using someone else's wifi signal in another country you should probably ask someone familiar with the legal system there. In the USA it would probably be considered illegal. In my opinion, yes, you should have been asked first.
Yes, it could downgrade your network, especially if many others are using the system.
It is also at least possible they could monitor your web traffic. Try to use secure servers only (the URL will beging with https rather than http) for any transactions you need to keep secure.
Best of luck.
Thank you so much for this information.
I can't answer the more technical questions in your second paragraph yet, but will investigate further and maybe take it up with my web provider.
Thanks again for taking the time to answer,
All the best