With regards to (4), it's been well known for a while that since Social Security doesn't check the payments being made into the program with any sort of scrutiny illegal immigrants can often get away with giving the social security numbers of dead people to their employers. Here's an article from 2024 that mentions the problem.

https://bipartisanpolicy.org/explainer/immigration-social-se...

From a policy perspective making it harder for illegal immigrants to be employed might make it worth cracking down on this. But doing so would cost the government money both by preventing these payments into Social Security that don't have to be paid out and also the cost of the crackdown itself.