That depends on how we define reconstruct. Lets see if I have this right. The chromosomes split into two pieces. From each of these two pieces (really each piece seems to be multiple pieces, but the term piece refering to each complete half of the DNA should suffice) the entire and correct dna sequence can be constructed. Since the DNA was once whole, reconstruct is appropriate.
Ok so far, the term you're looking for is "replication"; both strands of the DNA molecule are seperated from each other by enzymes and are used as templates to attach complimentary nucelotides; the end result is two copies of the same molecule, where before you had one. This occurs during prophase of mitosis.
At the end of this phase, there are two complete copies of the DNA of one cell. These two copies go their separate ways and become two cells.
So far, so good. Your wording earlier confused me; I thought you were referring to the "reconstruction" of the paired chromosomal copies that seperate in telophase of mitosis.
What is the name of the step where the separated stands of DNA gather together the parts needed to make itself whole again.
That's still prophase; to correct your misapprehension about the process of DNA replication, the separation of strands and the development of the complimentary strand occur in one pass along the chromosome, simultaneously. They're not seperate steps in mitosis/meiosis.
And back to one of my earlier points, what is the name of the process when the half DNA of the sperm and the half DNA of the egg (each formed my mitosis) combine together to complete conception?
That
is conception. Again, to correct what appears to be your misapprehension, gametes do not contain "half DNA"; they contain full chromosomes with complete double-helix DNA. They simply contain half of the
chromosomes of a normal somatic cell.
During conception, chromosomes from the egg and from the sperm combine in the egg and a nucelus forms around them, but the chromosomes do not physically attach to each other or anything, they don't combine in the way that one strand of DNA is combined with its compliment to form one chromosome.
All that happens is that the 23 chromosomes from the sperm are dumped in with the 23 from the egg, and a nuclear membrane forms around them.