It depends on the type of web development you're doing. If you're doing client work where the hours are fairly predictable, then absolutely. If you're at all involved in sysadmin stuff, where you need to be able to restart / configure servers when they go down at odd hours, you'll need some internet access outside of the library's working hours.
Public libraries are one of the greatest public works in our society. I'd love to see more web entrepreneurs utilize them.
Absolutely! You'll want an online development environment. I recommend Cloud 9 (c9.io) for basic stuff and Nitrous (https://www.nitrous.io/) for more involved projects.
Presumably the comment you're replying to is assuming the OP will be using a library computer as well, in which case an online development environment would be preferable.
Public libraries are one of the greatest public works in our society. I'd love to see more web entrepreneurs utilize them.