Don't go buying a Core2Duo, way too outdated. Just a motherboard/cooler for the processor you already have might be ok, but I wouldn't recommend it.
You can definitely stream on a $500 budget, if you don't need windows/case. Is it worth it for the $? Well up to you. I'm feeling like window shopping today, so here you go:
Common to all options:
HDD - 500gb Seagate Barracuda 7200rpm ($75) - there's also a 1TB version for $100 if you want more space
Case - you have one ($0)
PSU - Rosewill Green 530 ($50) - not a lot of good deals on lower-wattage quality power that I saw at the moment. This one is decent enough, quiet enough, and cheap enough, though not the best all categories
Option #1 - the simple all-rounder
using a non-overclocked intel quad-core for decent performance streaming & all round flexibility + a video card that can handle games more demanding than SC2 & Dota2 @ at least medium settings (S2 & Dota2 at maxxed graphics settings) for when your friend wants to load up Skyrim/branch out to new games. This CPU should also allow a smooth med-high quality SC2 stream. No USB 3.0 or SATA III, but those are unnecessary luxuries.
CPU - i5-2400 ($185)
CPU Cooler - Intel Stock Cooler
Mobo - Zotac H61 Cheapo ($50) Running Sandy Bridge on stock is easy as pie for budget mobos. (Note: not sure if this mobo has 2 controllable fan headers, may just have 1 controllable, 1 3-pin. If you need 2 pwm headers, double-check or get a cheap Asrock H61.)
RAM - 1x4GB 1333mhz RAM ($18) - 2x2gb would perform a bit better, but this gives you the option of adding a second 4gb stick if you want more RAM later. See option #3 for the 2x2gb kit to buy if you want the small increase in performance now.
Video Card - Gigabyte 6770 ($95 AMIR)
Total option #1 - $490 AMIR ($510 - $20 rebate)
Option #2 - the streaming specialist
using an overclocked Phenom II hex-core, you'll have 2 cores to play SC2 on, and 4 cores to dedicate to streaming. The video card should run SC2 and Dota2 fairly well, but will choke on graphically demanding games at anything other than very low settings. This is the option for streaming in a high quality while still experiencing smooth gameplay. if streaming isn't a priority, may not be the best option.
CPU - Phenom II x6 1045T ($150)
CPU Cooler - Coolermaster 212+ evo ($30) cheap, good performance
Mobo - (~$100-$140?) Some good AMD overclocking mobo... sorry, my hardware knowledge isn't good enough here, I haven't purchased a mobo for this kind of processor. Ask around
RAM - 3x2GB 1600mhz RAM ($40)
Video Card - Radeon 6570 ($50 AMIR)
Total option #2 - $512 AMIR, may be more or slightly less (mobo cost)
Option #3 - the focused budget option
using a dual-core processor to lower the cost & not increasing the GPU cost for more demanding games, this computer will run SC2 (& moba games) on maximum settings with ease, is cheap, will provide ok performance for other modern games, but is most definitely not capable of streaming. You can increase the price of this option by moving up to an i3 (adds hyperthreading, the SB Pentiums are dual-core no hyperthreading) or a significantly more powerful video card like a 6870, but for your stated purposes, it's not necessary.
CPU - Pentium g850 + DVD combo ($95)
RAM - 2x2gb 1333mhz ($20)
MOBO, CPU Cooler, and Graphics card from option #1
Remove the DVD-burner from the general parts list (combo savings)
Total option #3 - $385
General note - normally I recommend the 6870 for budget gaming builds, since it provides real graphics power. But SC2 & mobas don't need real graphics power, so these recommendations are tailored to the purposes you stated - make sure that's really the purposes
.Edited by MisterFred - 3/30/12 at 2:55pm