For the longest time, I have wondered what our world would be like if all countries used Edison’s DC power distribution system for most lighting and appliances instead of George Westinghouse’s AC power distribution system?

From what I remember reading, Edison's system was unable to transmit power over long distances. 
So there would have been lots of small power stations dotted all over the place. 
My mind might be playing tricks on me, so if anyone knows differently, jump in. 

Their already starting to use it?, I believe the latest power distribution network idea is 800 KV DC
https://youtu.be/u4m84EvDUtc

For the longest time, I have wondered what our world would be like if all countries used Edison’s DC power distribution system for most lighting and appliances instead of George Westinghouse’s AC power distribution system?

A slight clarification, AC power was Teslas idea, George Westinghouse only funded it 

'The problem of transmitting electricity over longer distances became a recognized engineering roadblock to electric power distribution, with many less-than-satisfactory solutions tested by lighting companies. But the mid-1880s saw a breakthrough with the development of functional transformers that allowed AC power to be "stepped up" to much a higher voltage for transmission, then dropped down to a lower voltage near the end user. Compared to direct current, AC had much cheaper transmission costs and greater economies of scale — with large AC generating plants capable of supplying whole cities and regions — which lead to the use of AC spreading rapidly. '

That's what they said in this article about how it was done in the early days, and that's what I thought the OP meant. 

Before the grid existed, street lighting using tungsten lamps in the UK were run from large PETTER oil engine’s 
I have a book on the history of PETTER engines, and back in the early early days, towns would have 3 street lighting engineers, two guys would start/run and look after the engine when it went dark back at the engine/generator house, and another would cycle around town looking after the lamps and changing them if needed according to the book!
Imagine what a fantastic job that would of been?

There are modern HVDC systems in service, which was unfeasible during Edison's time due to the lack of the required power electronics in order to convert DC current to voltages suitable for long distance transmission and then back to a voltage suitable for end users.

IIRC there were small remnant DC power distribution systems still kept for specialized individual use that survived long after most places switched to AC. In New York, the last of the DC distribution network was only shut down in 2007. I suppose buildings still requiring DC will have on site conversion now.

Another example are cities with older electrically operated transit. In Toronto there's a set of substations that convert the incoming AC power to a 600V DC system in order to power the streetcars and the subway trains in the city. 

Definitely what was destined to win was any system that allowed to transform the voltage tothe level best suitable for the distance and power levels to be transferred. The DC may make the transmission line at a given dielectric strength (so max peak voltage) and resistance (wire thickness,...) more efficient and/or able to transfer larger power at the given losses, but the ability to transform in a reliable and efficient way the power between the voltage levels that leads to the cheapest and most efficient line design for the given power is still way more important. And because at that time the DC offered only rather inefficient and maintenance very intensive rotary converters, while AC had way more efficient, more reliable and less mai tenance intensive transformers, it was the AC to win. The motor simplicity was a factor also, although it meant dealing with a 3-phase power in machines or installations (1-phase motors are of similar complexity and lower efficiency as DC), so over all I don't think motor construction was that much decission factor.

The modern DC systems became possible only after efficient power switching elements to construct the rectifiers and inverters became available.

The question may be, what may have happened is if it somehow (hypothetically) happened that the modern power electronic was available at the time of the "wars of currents", whether the ability to turn DC to AC for a voltage conversion and back to DC with cheap and efficient technology allowing the higher efficiency of both the DC power lines, as well as transformers operating at a more optimal frequency for the given power level, won't be a good motivator to stay with DC.