Can the arduino ethernet sheild/library perform these functions?

Hello all, so I want to toss together a mini ethernet tester of sorts, not for testing cables, but checking to see if the network is up, and what sort of access the device has.

Plan would be to use one of these modules as they are cheap and small, so it could fit a compact case.

I basically just want to plug it in and then display various variables, and I'm not sure if it can based on the arduino ethernet libraries I've taken a peek at.

Essentially I want to have it display the default gateway, subnet mask, it's own IP address, if the network is using ipv4 or ipv6 (though that's not really super important), and then also ping out to external ip addresses/sites, and then if it gets a reply, say yes or no.

Another neat thing to do would maybe be use one of these web server modules, if it can do the same, and have it display a little text page that you can try and access.

So, I'm wondering if it's possible to do this with some of the arduino libraries, and if this hardware would be appropriate.

edit: apparently that module does not work with switches that are capable of gigabit speeds, so you need something based on the w5100 like this, from one post I found.