And yet, where I'm from(Poland) it's extremely common to have the DPF removed from a new diesel car, pretty much any workshop will do it and fit a DPF emulator instead so the computer doesn't complain. All of that to avoid a potential cost few years down the line to have the filter cleaned/replaced. I'm utterly disgusted by how common this behaviour is.
I wouldn't say it's very popular... I rarely see trucks that are sporting DPF deletes, though I guess if you live somewhere like Texas with an insane truck culture, it's probably more popular than elsewhere.
No, to my knowledge you can pass the annual technical test without any issues even with the DPF filter removed, unless the engine has other issues and is literally emitting black smoke - the particulate emissions without the filter are not enough to trigger a fail(essentially the threshold is so fantastically high that modern diesels are far below it, dpf filter or not).
Honestly I've never heard of DPF removals before (edit: reason being mandatory car inspection usually finds this modification where I live, apparently it used to be a thing for some time and then the inspection protocol was changed). I only heard about people bypassing the AGR (exhaust recirculation), but that's independent of diesel/petrol (both have them nowadays).
That being said DPF issues seem to mostly impact people who only use the car for short distances; I've lived with DPF diesel cars for ~10 years and no issues. That being said any ICE powered vehicle is completely unsuited for travelling short distances, not just from an ecology point of view. Some more than others, though, it seems.