Every area is different as far as norms and people trying to cut you off, etc., but I've never had an issue with people cutting me off with even more space than that in Denver, Houston, Los Angeles, Seattle traffic.
You should always be able to see the bottom of the tires of the car in front of you from your position in the driver's seat, as it reduces the likelihood of an accidental collision due to inattention, reduces the severity of pileups if someone rear ends you and you are on top of the guy in front of you, and, from my perspective as someone who rode motorcycles for years, it means you always have enough room to cut your wheel and get out of the lane you are in if something unforeseen happens.
Yeah - not around here. Here people will cutt you off if you leave 2" of space (figuratively speaking). And yes - I sometimes have been one of those people.
On the "You should always be able to see the bottom of the tires of the car in front of you" - this is one of those things like "you should always be nice to baby seals". Nice - but not really applicable in real life. I've been driving for almost 20 years here in the US - in this bumper-to-bumper - and more before that in Europe. I've never rear-ended anybody despite driving inches behind a lot of the time. It is simply a matter of reflexes - people with better reflexes can do that. Maybe most can't - but that has nothing to do with me. (And yes at speed I normally wouldn't tailgate - with the exception of when somebody is blocking the left lane and really needs to get out of the way).
Edit: Speaking of seeing bottoms - and somewhat unrelated - another thing I don't like about the Model 3 is the poor rear visibility from the inside mirror. The back of the Model 3 is too high - so if there is a car right behind you - particularly at night - it's very hard to see (as its headlights are hidden below). In my BMW I can always see what's immediately behind me.