how to put the shoutmix under the community activity on homepage? where to put the code?

hmmm.. maybe in one of the ad boxes?
GOTO: Admin Panel > Themes > Blocks
i haven't tried dropping complete embed codes into those yet, but it SHOULD work.

if you're comfortable with a text editor, you can paste it directly into your themes template (page.tpl.php)
just be sure to make a back-up copy before you start.. just in case something screws up ;)

good luck!