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If true, the why were early railroads built as narrow as 3 feet and as wide as 5?

Simple money vs capacity. The narrow gauge railroads are much less expensive to build. Especially in mountains where the ability to make tighter curves can reduce the cost of building 3' gauge can be less than half of standard gauge. Going in the opposite direction the wider the gauge the greater the capacity and better 5the ride. If you had the cash building wider would make for a faster railroad. I K Burnel built the Great Western to 7' gauge and was running 60 mph average speed trains in the 1840s.

Why the standard. Because when Stevenson built his railroads he changed as little as possible. He didn't give the gauge a second thought. The jigs existed for building 4' 8.5" freight cars and he saw no reason to change what already worked. But he built a lot of track mileage. So when interchange of cars became a major issue people who could interchange with Stevenson's lines had a big competitive advantage. Eventually even the Great Western converted to 4-8.5 in order to be able to interchange with the other railroads.

In the US the norther states started off buying British locomotives so adopted their gauge because buying an off the shelf locomotive design was cheaper. In the South the use 5' gauge. But due to a guy named Sherman most of those lines and especially their locomotives got trashed and it was Northern engineers that rebuilt them. The changed the gauge so they could use the locomotives and cars they had brought with them from up north.

Interestingly in Japan and India the "Standard" gauge is 3' 6" and in Russia the standard gauge is 5". In the case of Japan and India it was because their railroads were laid out using the economical British narrow gauge. Russia's first railroads were laid out by engineers from the Southern US so they used the gauge they were used to.

The lesson to be learned here is people usually use what is to hand and don't give it a second though. If you are building a house are you going to design a custom screw for holding up the drywall or just use what you can pick up at Home Depot? The only time gauge on railroads in the US was given serious consideration was when the D&RGW was laid out. They needed to build it fast and the were desperately short on cash. So they built 3' gauge and scaled the equipment down to match the smaller gauge. They started over from scratch because it was the only way to get the railroad built.

I recall that among the advantages which the North had in the American Civil War were that (a) railroads connected at central terminals in cities and (b) they all used the same gauge.

In the South, on the other hand, each railroad would have its own terminal in a city. Railroad A might have its terminal on the east side of (Stonewall) Jacksonvania while Railroad B would have its terminal on the south side of the place. Goods would have to be hauled by wagon or barge from one to the other when transferring between one railroads. It is my understanding (and I’m not an expert) that southern railroads did not have a standard gauge, so that even if the railroads were connected, the locomotives and rolling stock could not run on the other line.

in Russia the standard gauge is 5".

Wow, that's really narrow! ≤}B^)

I find the 2-foot Ffestionog & Welsh Highlands roads especially charming. Seems right out of Dickens--or Harry Potter.

in Russia the standard gauge is 5

Also, in the US the ends of each rail is alternated (left, right, left, etc) and produce the familiar "clickity clack" sound. The ends of Russian rails are simultaneous and produce an endless series of thuds.