Broken stem tube – LBS "just" lowered handlebars. Is this safe? I have asked this elsewhere on the internet as well, but this is the Stack Exchange adaptation.
I have three related questions and I'll try to make it clear what they are. Here's some background where I've tried to boldface the most important parts:

Very recently I bought an old 1976 steel racing bicycle from the local Craigslist-like site. The previous owner said pretty much only the frame was original, and most everything else had been replaced recently. Sure enough, everything looked new and from a quick inspection nothing seemed out of place either. On the short ride home that evening, it rode like a dream as well.
The next day I rode to and from work and noticed the handlebars creaking when I put weight on them. That continued to and from work today, so on my way home I popped into a local bike shop and while they didn't find anything immediately, I asked a question that prompted them to remove the threaded stem.
Apparently, the stem tube had broken clean across the &quot;slanted&quot; part. I've attempted to illustrate where the break was here:



Their best guess was that the previous owner had raised the handlebars too much, and the metal just couldn't cope with the load so it cracked. And when the two pieces of the stem tube ground against each other that made a creaking noise.
You can imagine how glad I was that I got it investigated.
Since the LBS didn't have a replacement stem in store, they decided to put the handlebars back in with the existing, broken, stem tube. They reasoned that if the handlebars are low enough, there should be enough stem tube still inside the headset that nothing bad will happen. They assured me that I would be safe as long as I don't attempt to raise the handlebars.

Question 1: I guess there's a reason the stem tube is &quot;slanted&quot; like that. Why? Does it matter if my stem tube is broken and not slanted anymore? Do I risk breaking more things? Am I really as safe as the LBS wanted to assure me? (Based on the reading I've done, I'm guessing the slant is there to make sure the steerer turns with the stem, and the LBS might be thinking that the friction between stem and steerer should still accomplish that.)
If I wanted to, they said I could call them and they might get a new stem and do the replacement. Their best estimate was something like $70 dollars for the stem and work to replace it.
Now, I have some bike maintenance experience – but mostly the basics. I've intentionally refused to do anything that requires a torque wrench. However, I made sure to watch them work on my headset now, so I'm fairly confident in my understanding of how everything fits together. I think I may be able to replace the stem myself.
Question 2: The LBS guessed it would be about $40 to get the stem, but also said they had no idea where to order such a thing because it's a pretty old bicycle. Is there any chance I can get a hold of it for cheaper?
Question 3: If I think I can replace the stem myself, am I deluding myself? Is it hard? Easy? What might I have missed when looking at them work on the headset of my bike?
I guess those are the most pressing questions, but I'm also sure more will pop up. Thank you for taking the time to read!
Here's how I would fix the problem.  This could all be done by somebody with basic tools and basic knowledge.  

Get a quill-threadless stem adapter and a new stem. This will allow you to fit the new stem to the handlebars without removing anything from the bars. The old stem can be removed destructively by removing the cinch bolt and prying the thing apart since it is already broken and no longer fit for use. 

Using a new theadless stem will give you a huge variety of stems to choose from so you can find the right length and angle to fit you comfortably and match your riding style.

I'm not recommending those specific products, but those are the type of products you should be looking for. Make sure the handlebar end of the stem is the proper size for the bars you already own.

I got a threadless stem adapter a couple years back for my old bike and it makes working on anything to do with the handlebars so much easier.