This all seems very strange to me for a couple of reasons. 64 bit Ubuntu still isn't "recommended" when downloading, but I've only ever seen it recommended these last years and haven't run into these problems since perhaps three years back or so with modern software. Steam is still in beta, but I'm surprised at why it shouldn't be immediately installable for 64 bit. I didn't have any problems at all installing this on my 64 bit Arch Linux (though I don't remember if it had to pull some extra libraries during installation, it's very possible - but it's an "unsupported" distribution so why should it be easier than a "supported" one, sigh...
From my experience
apt-get can be slow, but it shouldn't be a crawl if your computer is even reasonably modern.
I'm honestly not a big fan of recommending people to blindly copy and paste stuff into the terminal. Since I haven't done Steam on Ubuntu I can't help with that (hopefully somebody in this thread has, though!), but to get you sort of up to speed on what you're doing:
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dpkg is an installer for .deb packages. Like the poster said, you can apparently instruct this to accept i386 packages instead of only x86-64 by the command posted above. Never had to do that myself.
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apt-get is a program which gets .deb packages from repositories and (I think?) then calls dpkg to install them.
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apt-get update will update the available package lists in the repositories.
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apt-get install foobarwill, of course, install the given package
foobar if available.
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sudo gives root privilegies to a command.
Basically, whenever I run into install problems I first make sure that all of my installed packages are up to date by running
sudo apt-get update to update the list of available packages and versions, then
sudo apt-get upgrade to upgrade all out-of-date packages.
There are GUI frontends for this, but they just call these commands. Have you made sure that everything is updated before doing this? Doing that solves most of my installation issues whenever I have them.
That's about the only advice I can give right now, except that if you're really dying to try this out to download and install the 32 bit version of Ubuntu instead.
Finally, do keep in mind that it's still in beta
That said I'm very surprised that these problems occur. Hope you find a solution!