Nope. Only certain PSP models are capable of doing this I believe the PSP 2000 and PSP 3000 are the only models who can do this. As far as I'm aware the PS Vita isn't capable of doing such a thing, at least not with the current model.
My friend had a cable, and we played the PSP on the TV. It was really cool actually. I am hoping the Vita can as well because I actually want to pick one up now on eBay or something because Sony said don't expect a price drop any time soon.
You can use a component cable to hook your PSP to the tv. Unfortunately, without a plug-in, it will cause blackboxing. Even if you get a plug-in to maximize your tv's resolution, the graphics will degrade in quality due to the screen being stretched out. Only a few games look good while doing this.
No, but I have looked up some videos recently on it. As already stated it doesn't seem to do well on a big screen. Also, I'm not sure it's entirely accurate based on some of the comments, but it doesn't seem to be able to play games on screen or there was some catch to it. Can't remember. Anyways it doesn't feel worth it though it might be fun to try at least.