1) I honestly don't know if this an actual router or not. I said it was one because it allows devices to connect to it.
Seagate Wireless Plus creates its own Wi-Fi network, just as a wireless router does. There's no network connection upstream from Wireless Plus, so it's not a router in that regard.
Airlines and equipment manufacturers had to go through an FAA certification process for in-flight Wi-Fi networks, proving that specific equipment would not interfere with specific aircraft avionics.
I suppose it's possible that passengers can now establish their own in-flight Wi-Fi networks with any equipment they bring on board.
Hawaiian Airlines has a very clear list of approved PEDs:
http://help.hawaiianairlines.com/app/answers/detail/a_id/2091
Rules for some devices might be different on some other airlines, but since the rules all stem form the same safety concerns and FAA regulations, airlines should be similar.
"Mobile Device Storage with Built-In Wi-Fi Streaming" is not in the list of Approved Portable Electronic Devices.
The list of Electronic Devices that are NOT Permitted for Use includes "Wireless devices (keyboards, mouse, etc.)." The Seagate Wireless Plus could be considered a wireless device, although it's a WiFi device, not a BlueTooth device.
Also, radio transmitters are prohibited. Technically, a WiFi transmitter is a radio transmitter, but probably not in the sense of the list of prohibited devices (CD radio, Ham radio, etc.), where such prohibited devices transmit signals for a long distance.
My Google searches did not turn up any recent pages banning devices that establish in-flight WiFi networks. Also, laptops can be configured to serve as WiFi hotspots, and there doesn't seem to be any concern about this.
Personal Electronic Device (PED) rules have been relaxed in recent years. A few years ago, American Airlines prohibited the use of USB-attached external drives. Now most airlines, with FAA approval, even allow the use of smaller PEDS during take-off and landing.
In short, the Seagate Wireless Plus seems to fall in a gray area. Although wireless devices and radio transmitters are prohibited, and although the Seagate Wireless Plus technically falls in both categories, I actually agree that Seagate Wireless Plus has more in common with a laptop than with the two banned categories. However, if you asked an airline ahead of time, it's likely the answer would be no.
2) IIRC, GoGo also charges just for using it. I just looked it up... $14.00/day/device (or $50/month). So add that cost.
GoGo Internet pricing is for Internet access (web, email, but not Skype or streaming services). $14 per day buys 24-hour access across multiple GoGo-equipped airlines. There are cheaper plans, such as for a single flight or even a single hour on a single airline.
For pay-per-view TV programs and movies, it is not necessary to buy Internet access. Also, a few websites are available without buying Internet access, such as AA.com while flying on American Airlines.