I remember the trick was to fly to another place and re-enter just a few days later to get another 3 month visa. Doesn't work anymore?

Depends on the country. It won't work in the Schengen because the visa waiver is 90 days within a 180 day stretch.

Not like anybody is checking this, I'm speaking from e experience. I had several coworkers over the years who used to live for much longer in Schengen countries without registration. In Austria this works moderately well so does it in Hungary, but of course it has some drawbacks too. Usually in countries where people tend to curb laws to their liking as well as "forget" to pay taxes (this is the case for Hungary) more often than not, this visa situation not a big deal

Yes, yes they are.

Enforcement may be discretionary in some regions, but like all arrangements like this, don't stake your life on such.

Depends on the country. Many weren't keeping digital records but most now are.

Some places started offering remote work visas:

https://expertvagabond.com/digital-nomad-work-visas/