They do have an ethernet cable, but it is fairly short, so you can use your pc on the beds. (The modem is hooked up to the little phone stand between the 2 beds, not at the main table or vanity table type mirror seating area.) We bought our own long ethernet cable, so that we could have our laptop on the table. You can also use a pocket router in your room to obtain wireless internet, but it is complicated to set up, and you would want to make sure you set the right security so as to not have other guests logging in to your wireless router setup. (DLink makes a nice pocket router, good for travel, but can be tricky for a novice to set up correctly.) The wired connection worked well.
The in room safe in the premium rooms was great to fit our laptop, cameras, etc.
And yes, the internet is free.