HVDC, despite the advances in technology, is still a very niche system.
The cost of the inverters is immense - so HVDC is only viable where the lines are of exceptional length and capacity; or where 2 independent (i.e. not synchronised) AC grids must be connected. The breakeven point is roughly 1000-2000 miles of length. For shorter legs than that, AC is significantly cheaper.
Pretty much this. While advances in power electronics has made rectifier/inverters somewhat cheaper, it hasn't really changed the breakeven point for DC transmission very much.
It is true that a conductor insulated to withstand a certain maximum voltage will deliver more power using DC simply because the DC voltage can stay (nearly) at that maximum value all the time. This can mean lower right-of-way costs and also contributes to lower line losses which explains why it can make sense to use DC transmission for long distances. (So does the fact that there are obviously no reactive power requirements.)
As others have pointed out, the ability to use transformers to raise and lower voltage (and thereby minimize power losses over distance) gives AC a huge advantage. Just as important, AC motors are much cheaper and more reliable than DC counterparts. Sure, you can use a DC motor to run a compressor or a wash machine; but you'd be silly to want to.
People have been proposing DC circuit breaker designs for decades, but I'm not aware of any actually in service at transmission voltage levels (although perhaps one or two have slipped by me). In any event, DC breakers are perhaps possible but far from being proven as practicable.
The big advantage to AC for switching and circuit breakers is that by AC's very nature the current passes through zero 120 times a second. As the current passes through zero, the arching current across the opening contacts is extinguished. If we can clear the ionized atoms from between the contacts quick enough, then the insulating strength of the material between the contacts may be enough to prevent the current from restriking (in the reserve direction) as voltage builds toward the peak value. Doing that is what AC circuit breaker design is all about. You don't get any natural current zeros with DC (and voltage is always high too). That's what makes DC switching so challenging!
I think we will eventually see homes built with a single central DC rectifier that feeds a parallel set of DC outlets throughout the house, which should be more energy efficient than having scores of small rectifiers in all of our electronic devices. Now if we can just get all the manufacturers to agree on some industry standards...