I never thought about the induction problems but you're certainly right!  We need to convert our HV systems to DC like the Scandinavians.  That would solve the induction problem though I'm not sure what method they use to step the voltage down.  500KV dc to 20 kV dc converters tend to be more expensive than transformers - though with current price of copper that may be changing.
There is no DC grid in scandinavia. There are multiple high voltage DC interconnects where grids need to be connected together across longer distances across water. And there are DC intreconnects (across water) with unsyncronized grids. And former Asea now the Swedish/Swiss ABB were a pioneer for this technology and still is one of the best companies for HVDC eqipment.

The step up and step down is made in transformers as AC.

There might be a small start of a kind of HVDC grid within in Sweden since a fairly long 400 kV AC line intended to add redundance to southern Sweden might get built as HVDC. I think this would be a good idea for technology demonstration but a bad idea for added redundacy per $ since a traditional 400 kV AC line easily and fairly cheaply can be connected to older lines crossing it between the endpoints giving extra redundancy. On the other hand it would be built with newer technology making it possible for it to convert active power to reactive power wich southern Sweden need more of.

The limit for AC thru cables is the parallell capacitance. Basically when the cable gets too long all the current fed into it charges and discharges this capacitance and no power is left to deliver in the other end.

If one would build a complete grid from scratch in the near future it might make sence to use +/- 10 kV DC or so to each and every house allowing cables with very little metal and losses and then step down the voltage. This will make more and more sence as power semiconductors are developed further. It has actually been sugested by some innovators in Sweden as an economical way to build a parallell redundant emergency power grid connected to centraliced accumulators and emergency diesels instead of every other company buing their own set that often is too small and badly maintained. On the other hand it would still be sensitive to lightening. And for the apocalypticons, EMP. ;)