Advice on buying a laptop (Please help)

Hello. I'm 15 and I live in Brazil, but i'll be traveling to the US this february (more specifically to Ft. Lauderdale and Miami), and I pretend to buy a laptop. Could you guys please help me?
First, I need to state that it it almost mandatory that it is under USD 500,00 , because otherwise I'd have to pay some taxes to bring it here.
Alright, so now my needs:

I'll be using it as a home computer, which will be almost always plugged in. I need it to have an HDMI output, as i'll plug it into my TV. I don't care a lot for the size of the screen, but I think under 14" is very small.
I'll use it mostly for web browsing, downloading stuff (a lot!), video playing (mostly YouTube) and some light gaming, like a GBA emulator, some internet games and maybe World of Warcraft, or something like this.
I was thinking on buying one with an i5, which is what I think is necessary, but I would like to know if an i3 is alright for it. The reason is that the only computers I found like this with an i5 were some ASUS, Gateways and Acers. For the same price but with an i3 I could get an HP, maybe a Vaio and so on.
I don't have the option of buying a desktop, as it wouldn't fit in my luggage   The prices here in Brazil are tremendously expensive, so this is why I need to buy there.
I'd like to state that my computer right now is an HP Pavillion dv2000, with an AMD turion 64 x2. I really hate this laptop, because it gets REALLY hot, it's very slow and the battery lasts like 30 min. Also, the CD drive and the SD card slot don't work anymore. This means I'd like a computer which runs cool, and that ANY computer I buy will be better that my own.
Sorry for the long post, and I'd really appreciate if someone read this and could give me some advice. Thanks!!!


Note: I've been to the US more than 10 times, so I'm not unfamiliar to the prices/stores, and me and my family don't have any problems communicating in english.

edit: I'm sorry I didn't explain it well. I thought about the i5 because there's one thing is mandatory for me: The laptop needs to be FAST. For turning on and off, for opening applications and for internet pages. Mine is really slow, and I'm traumatized with this. Do you think the i3 is good enough for me?