I can't reproduce the build failures reported here even with dpkg-buildpackage -A. However, I am going to add the discard-port proxies to d/rules that pybuild normally adds by default (this package doesn't use pybuild). That at least will prevent the tests from *actually* hitting the internet. I'll close #830281 against that change.
Yes, this makes the test suite fail during package build. I have reported this upstream at https://github.com/pycurl/pycurl/issues/424 However note that d/rules currently discards test suite errors, so they don't prevent the build from succeeding. This makes running the tests semi-useless but I think until upstream fixes their test suite, it's the best we can do. Therefore I'm going to tag this bug as unreproducible and reduce the severity. I also can't reproduce the C locale problem, but I'm going to consider that a separate bug so please file a new one on that issue.