usually 100mbps is the recommended for an online game.
I'm going to go ahead and say it depends on your game. 100mbps is certainly incorrect, unless that was supposed to be the speed of your network card. A T3 line, which is $6000 to $12000 per month, can hit 45mbps
For a game such as World of Warcraft, I played that over uncompressed dial-up.
A game such as FFXIV which requires a lot of downloaded information you may need a 10mbps connection for it to be very fast, but it would most likely run well on a 1.5mbps DSL link.
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yes of course you do if you are going to be online you need internet
Broadband
You need an internet connection to play online only
Online Enabled means that you can play >over the internet< with other people. Note that you need to be connected to the internet to play online.
There are lots of scams on the internet for free gaming pcs. Don't listen to them because they will most likely give you viruses and make you need another regular PC. The only way I've heard of actually getting a free gaming PC is by winning gaming competitions.