I would recommend looking to setting up a Joomla install (one-click install though the panel) and using one of the Streming Media extensions for Joomla (all free). I'm not overly familiar with joomla, but I'm sure it's possible to allow public uploads and then one of those plugins to make it streemable.
You'll probably also want to set the php upload limit higher than the current 7mb or your videos may be limited. Check out the Wiki Article on PHP.ini to adapt the server's install of php to read on your settings fine for your domain.
Dreamhost provides you with SSH access to your own account. You mentioned SSH root access, which is super-user access and Dreamhost does not provide that. You'd have to go and get a dedicated server from an other company to have root access.
However, I'm willing to guess that you don't really need root access to do what you want. Most things can be easily installed into your own user space.
Dreamhost does not support or help you install 3-rd part apps, except for the one-click installs. But even for those, once you've got it installed they don't really support it. You're own your own to configure the software and keep it working. Dreamhost will keep their servers working, and if there's a problem they'll fix it, but what you do with your web space and any apps is your own to deal with.
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