Question on measuring bean density. How is the density typically measured. Density is mass per unit volume so I’m sure there’s a standard way to do it.
Well, you have machines that measure moisture and density, but those are expensive, I’d guess mostly for the moisture measuring part. In alternative, you can put your beans in a graduated cylinder, which isn’t the most precise method, but it gets the job done. Anyway, you can’t miss by that much as to get the roasting part wrong. The important thing is to pack your beans tightly into the cylinder. There’s also water displacement, but you don’t get that much more precision, as opposed to the beans you’ll have to throw away. The best method is always a no waste method. Just fill up a 100ml graduated cylinder, weigh it, and that’s it. If you measure 60 grams, that means the density is 0.6g per ml. Some very dense coffees will go up to or over 70, while lower density coffees might be under 60. Now, there will be people that are convinced you should pay over $1000 for a density and moisture metre, but just leave it be. The moisture measurements are taken by the importer before he sells you the coffee and, unless you’ve kept it scattered in open air, there’s little chance for a change. And if you take a density measurement that is 0.01g per ml off, you won’t become a better roaster if you take that measurement with a $1.5k metre.
Mill City talked about graduated cylinder method mentioned by @braca19452f9m in one of their videos on YT. Cheap way of doing it if you have a cylinder or willing to buy one (inexpensive) since likely you have a scale already.
New question regarding bean density. Does a higher density bean mean it has less moisture, or does moisture not play a role?
Moisture does not play a role in density, but it does play a role in roasting, although it will be rather difficult to find something that affects roasting that much, unless you buy some 2009 lot from somewhere. So, no, it does not mean that. As stated in my post above, if you’re not someone who imports large quantities of coffee, you should trust those that do and not worry too much about moisture. Or, expressed in the form of a title of one film, learn to stop worrying and love the bomb. 
Virtual Coffee Lab has a good video on how to measure coffee density at home but keep in mind that it is not industry standard. However, it gives you a good idea on how it should be roasted.
I always buy greens from reputable buyers that share the moisture content, density, altitude, region/farm, etc…