Can you connect an External Display with a higher Refresh rate to a Razer Blade Laptop that has a 60hz Display

It's possible to do this electrically, although mechanically the two cables interfere. Microsoft sells a special version of the AV adapter that you can use to plug both cables in at the same time, or you can break the outer casing on the AV cable so that you can fit the HDMI cable in at the same time.

Note that plugging both cables in at the same time might cause some issues if you plan on switching between inputs on the monitor while your Xbox 360 is still on, see:

Why does my Xbox 360 reset when I change the input on my TV, unplug the HDMI cable, or turn the TV off?

Related to my answer to this question, there are several options for streaming live gameplay. Two noteable options would be Open Broadcaster and Xfire in combination with TwitchTV as shown in this video.

Now the big catch is that you are going to need a lot of hardware to get the output from your XBox connected to a PC that can use these applications. There is a pretty good walkthrough in this video.

There are two main ways to go about this:

1. Split the HDMI out of your XBox and route one channel to your TV and one to a PC.
   You will need:
   • An HDMI splitter
   • PC with an HDMI input, typically via an add in graphics adapter (only input needed). In the video they use an Avermedia card
   • 2 HDMI cables
2. Route your HDMI to a PC "capture card" like the Black Magic Intensity series.
   You will need:
   • The capture card of your choice (input and ouput)
   • A PC compatible with the capture card
   • 2 HDMI cables

I'm not sure which option is really cheaper or better. I personally have a Black Magic card, but I mostly just record videos from it. If you are sizing up a rig for doing some recording, do consider a fast SSD drive - as recording in high def requires fast storage access.