Correct me if I misunderstood . Nucleus of the cell act independently of the nucleus of mitochondria
Mitochondria don't have cell nuclei. Cell nuclei are only found in eucaryotic cells, the kind of cells in which one finds mitochondria. Protozoa, fungi, plants and animals are all comprised of eucaryotic cells. Just about all multi-cellular organisms consist of eucaryotic cells.
Mitochondria themselves more closely resemble procaryotic cells, whose representatives are bacteria and archaea. Procaryotic cells are smaller and simpler structurally (though not necessarily simpler biochemically) than eucaryotic cells. Many of the cellular structures ('organelles') found in eucaryotic cells, including a cell nucleus, aren't present in procaryotic cells, or in mitochondria.
https://en.wikipedia.org/wiki/Prokaryote
https://en.wikipedia.org/wiki/Eukaryote
The earliest life on earth seems to have been procaryotic. It seems to have appeared very early, perhaps right after the earth formed. Explaining that is one of the great mysteries of biology. For a long period, bacteria and archaea were apparently the only kind of life on earth. Then suddenly eucaryotic cells appeared. How, when and why that happened is another of biology's mysteries. Current speculation has it that the process that generated eucaryotic cells included (but wasn't restricted to) cells taking up residence inside other cells as endosymbionts. Mitochondria and plant chloroplasts (which also have their own DNA) are descendants of these.
Then at some point these more capable eucaryotic cells started to come together in multi-cellular organisms and the 'cambrian explosion' occurred, generating most of the body plans that we see in animals today. Coelenterates like starfish, a wide variety of worms both segmented and non-segmented, arthropods, chordates and molluscs appeared, all in a relatively short period of time. That's another of biology's mysteries. Though evidence now suggests that the explosion may have been due more to the evolution of hard body parts like shells that fossilize well than to all the familiar taxonomic categories of multicellular life appearing from single celled ancestors so quickly. There's growing evidence of an earlier period, prior to the 'cambrian explosion', when many early, experimental and in some cases abortive soft-bodied taxonomic types existed.
https://en.wikipedia.org/wiki/Ediacara_biota
The nucleus of mitochondria generate its own daughter units for energy production . So both unit act independently.
Mitochondria don't reproduce independently. Most of the proteins that make up the mitochondrion are encoded in the chromosomes of the larger surrounding cell's nuclear genome, and manufactured in the larger cell's endoplasmic reticulum and Golgi apparatus. (More of those eucaryotic 'organelles'.) So the larger surrounding cell reproduces its mitochondria, just as it reproduces its other organelles. At best, the mitochondrion assists in its own reproduction.
and the Nucleus and mitochondria have their own DNA.
Yes. But the mitochondrial DNA no longer has enough genes to fully reproduce a mitochondrion.
Question : does the mitochondria DNA have a double helix ( double strand ).
When people talk about it forming a ring in the same way that procaryotic bacterial DNA does, they aren't saying that it isn't a double helix. They are saying that the strand of double helix meets and fuses at its ends and forms a circle.
In eucaryotic cells, the double-stranded DNA is organized much more elaborately, into chromosomes.