I heard that the Romans never had the concept of zero nor a numeral to represent it in their system of mathematics. So, first, the number zero was invented by the Babylonians, but was never introduced to Rome?

Well, the Romans did, however, have a way to represent none: nulla, the Latin word for nothing. But, in modern days today, instead of nulla/I/II/III/etc, we write 0/I/II/III/etc. Zero is a more specific concept in math than "null" or "absent". I heard that no countries in Europe had zero until the 1200s. So, the Romans used less standardized concepts until the more specific concept was borrowed from somewhere else?

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    $\begingroup$ Romans did calculations with stones moved along grooves. ("Stone" is "calulus" in Latin.) After the calculation, you copy down the answer: 3 in the hundreds column and 6 in the units column ... CCCVI No need to show zero in the tens column. $\endgroup$ – Gerald Edgar Jan 4 at 13:34

The digit 0 was not needed in the Roman numeral system because this is not a positional system. The only case when it was used was when the number was actually zero, which they called nulla.
Roman digits have a fixed value independent of where they are in a number. For example, the letter X means 10 per se. Its value is added to the other digits in a number, with the exception that if it comes before a higher value digit it is subtracted instead of added, e.g. XL means 40 (50-10) and LX means 60 (50+10).
In the Indian or Arabic system however, each digit has a value which depends on its position, e.g. the digit 2 can have the value 2 if it is in the last position, 20 if it is in the next position, or 200, etc... In this positional system you need to fill every position in the number to preserve its meaning. That's why number zero becomes necessary to indicate an empty position.

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