I'm in the process of ordering my components for the gaming pc I'm building. I'll be using a single GeForce GTX 570 that has an mini HDMI and 2 DVI ports. I currently have a Insignia 32" 1080p 60Hz HDTV (http://insigniaproducts.com/products/televisions/NS-32L...) and I just wanted to make sure that I wouldn't run into any problems. My guess is that most PC monitors are 120Hz+ instead of my TV's 60Hz and I want to know if that will be noticeable at all. Should I look into selling the TV and purchasing a monitor? If I do keep the TV what would the best way to hook it up be? mini HDMI to HDMI cable? DVI to HDMI cable? Will there be any lag compared to a PC monitor? I currently have a MacBook Pro and have used a mini Display Port to HDMI cable to connect it to this TV but it seems to be a split second behind the built in MacBook Pro display. I'm guessing this is due to the mini display port to HDMI adapter or the fact that my graphics card is only a GeForce 9400M GT. My primary use for this PC will be gaming and I just want to make sure that everything will work as expected.
Most monitors that I see are still 60Hz. 120Hz is nice if you want to be able to benefit from fps higher than 60. For your TV are you using speakers or the tv's speakers? If you are using the TV's speakers then you will want to use the HDMI otherwise it shouldn't matter which one you connect to.
I'd try to steer clear of displayport adapters if you can. Passive displayport can use an adapter (sounds like what you were using), but active displayport requires an adapter with an external power supply (a usb cord for example).
I'd go HDMI => HDMI.
120Hz is available on computer monitors, but is by no means a common feature. The "best" high resolution, larger monitors are still 60Hz (27" + monitors greater than 1920 x 1200 res).