You can hook it up to a TV provided your graphics card has the right output. Most modern graphics cards have HDMI ports. My card has HDMI, VGA, and DVI. Personally, I have a 24" monitor that goes up to 4K resolution, so I use that for gaming, but I have hooked the PC to the TV many times before.
You can use a laptop or desktop, it just has to meet the requirements. I have a desktop I built especially for gaming. Technologically, it far exceeds a PS4 or Xbox One. You dont need to go that far and depending what you already have, you may meet the requirements already. Processor and graphics card are the most important pieces. If your PC does not meet the requirements now (find out by going here:
http://www.systemrequirementslab.com/cyri), you may be able to find simple upgrades cheaply.
Yes, you can use a 360 or even an Xbox One controller. I personally play FIFA with a regular ole 360 controller. A large majority of games released since 2009 or so all support them and Windows supports them natively since XP. The 360 controller is the #1 used PC controller. Xbox One controller support arrived recently in the form of drivers and I actually just ordered one today. I have close to 900 games spread out over Steam, Origin (EA's Steam, which is where FIFA is) and Uplay (Ubisofts Steam lol) and I'd say 60% of them have full controller support, meaning button icons on the screen etc.
Valve is releasing their "Steam Machines" very soon (just google Steam Machines) which are basically PCs meant to be treated like a console (living room etc), except it has the added functionality of a PC as well as a gaming console. They are trying to make it very easy for people to get into PC gaming.
You could almost assuredly build a gaming PC for $400 that at the very least meets the One/PS4s technology, if not exceeds it, plus its a PC, so it does a lot more than gaming even if its built for gaming.
I was a console adherent for a long time, but the PC is light years ahead and always has been. You can get some absolute steals in terms of software prices. Steam always has massive sales, plus they have a trading ability, which means you can get games much cheaper, even pre-orders. There's also game bundles, which are literally 5 games for a dollar (Google Humble Bundle and see some of the things they've done and they're far from the only bundle company). The one knock is that the consoles do get exclusives at times that I miss out on. I greatly miss the NHL series, but for the most part the majority of games that come out on console also come or will eventually come out on PC (i.e. Ryse is out on Steam next month and the PC is getting the "next gen" GTA V in January).