It may have changed recently but that's not quite how the restrictions worked on Wall St in my experience (8yrs in the front office at Goldman). There are lots of restrictions to the point where owning broad funds often becomes simpler, but you can do single-stock trades etc if you want to.
See another comment where I explain more but how it used to work for me was you have to trade through the employee trading desk (and aren't allowed to have unmonitored brokerage accounts) then any trade gets vetted against various lists (some but not all of which are available to you ahead of time to protect deal confidentiality) and then there are static criteria about how long you have to hold particular positions etc.
They could put investment money in like an independent index fund maybe...? Although honestly there should probably be a "Congressional Investment Fund" that they have to put any stock investment money into that invests equally across all publicly traded companies or something.
Sure, why not? If they aren't willing to sacrifice that for their constituents, as to reduce any risk of conflicts of interest... I'm not sure they should be in any position of power to begin with. Why not just have their savings/pension in a known, public index fund?
You could, but I'm positive that statisticians much smarter than me have devised a test tunable to any sensitivity one desires. The real question is, "What's our threshold of sensitivity, and what are the FPR and FNRs are we willing to endure?"