I strongly recommend Wordpress. You can download it for free from Wordpress.org.
Also, if your hosting company uses cpanel and Fantastico, you can install it directly from your control panel. I recommend you check with your host to find out if they have Fantastico or can get it.
And to answer your first question, yep, you can install the blog in the a /blog directory on your site. That's what I recommend doing, that way you are directing traffic to your domain.
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