You buy mice that are likely to develop it, for example at https://www.jax.org/mouse-search (I notice they offer a service to use CRISPR to design a mouse strain for you, too)

Not a stupid question. Mice (and all other animals besides maybe nonhuman primates) don't get Alzheimer's. There are mice bred to display symptoms of Alzheimer's. Because nobody knows the "root cause" it's a bit of a catch-22. The same holds true for many other neurodegenerative/psych disorders.

I don't know in detail, but suffice it to say that for many years we've had mouse strains with particular genetic alterations that appear to mimic Alzheimer's. Not my field, so google for more info.

Not a stupid question at all. The answer is the mice are genetically engineered to overexpress beta-amaloid (or sometimes Tau).