Github tool to validate links in markdown

I think what you want to use is awesome_bot.

It doesn't provide the badge you want but it does check URLs in files.

From what I get from your question, Travis is actually enough to do the checking task.

I have already implemented it in this project. It's based on nodejs package named grunt-deadlink, Travis-CI configuration is also included. Unfortunately it doesn't support nightly test (as far as I know).

For shiny badge you can simply use this badge generator service.