Why the word "mind" can be used as a verb, synonym of "pay attention to"? It has the same etymology of the "mind" (centre of thought, feelings, brain) noun? When it is better to use "mind" in place of "pay attention to" (or similar phrases)?
The verb, "to mind" is descended from the Old English noun "gemynd", which means "thought", "memory", "thinking", "commemoration", and "intention". This word itself is descended from the hypothetical Proto-Indo-European word "*men" meaning "to think", "to remember", or "to have one's mind aroused".
The most interesting thing (for me) is that this word has cognates in Sanskrit, Latin, Ancient Greek and Russian.
I guess the word would have originally been used in the sense "to remember". E.g. when one is saying "mind the kids!" what they really mean is "don't forget the kids!". The word, with usage, would have eventually gained its meaning of "to pay attention to", simply through its continual usage in that sense. It gained this particular meaning ("to take care of", "look after") round about the 1690s.
Seeing as "mind" has so many meanings to do with thinking, remembering, as well as the whole idea of spiritual embodiment of one's thoughts, it only makes sense that it would have so many uses with different flavours of meaning today.
As for exactly how it happened; that would require a lot more research, and possibly a time machine.
For usage, I don't think it makes much difference whether someone were to say/write "pay attention to the kids" or "mind the kids". The latter quote is a lot shorter, but also has a slightly different meaning.
Warning: Opinion follows: "Pay attention" implies that you need to watch them, but doesn't necessarily mean you must look after them as well - although it usually would. "Mind the kids", implies that you must watch the kids and keep them out of danger - though if anyone were watching kids, and didn't take care to keep them out of danger, they would be a right drongo.
"Pay attention to..." can also mean "Pay especial attention to...", for instance, in the sentence
It is obvious that the listener should look after all the kids, with special care regarding Sally, as she may be prone to being more naughty or adventerous, and likely to get into trouble/injure herself.
To put the two phrases in the other order:
Requires the extra adverb to bring the same meaning to the table.
"Mind" is short for "keep in mind." In this context, it is therefore synonymous with (but not the same as) "pay attention to."