Some simple answers to most of your questions, followed by a detailed answer to one of them.
The DNA nucleotides are free, floating around in the nucleus (not in the cytoplasm as the other person wrongly stated).
The bases "stick" to one another via hydrogen bondings.
There is no extra backbone: the backbone of a DNA strand is a repeating sugar-phosophate pattern. Each incoming nucleotide already has a single suger-phosphate unit in it: adding the new nucleotide extends the sugar-phosphate backbone by one unit.
There is no "who" in the cell. It's all chemistry and physics. Here is an explanation of how the cell transitions from the G1 phase into the S phase.
Cyclins
As their name suggests, cyclins are a group of proteins whose concentrations repeatedly “cycle” up and down over the course of multiple rounds of division (one “up” and one “down” per cell cycle).
Cdks
Cdks (cyclin-dependent kinases) are protein kinases that require binding of cyclins in order to be activated (a protein kinase is a protein that phosphorylates other proteins). Their concentration remains relatively constant throughout cell division.
Cdk1 (Cdc2)
Cdk1 (cyclin-dependent kinase 1) and Cdc2 (cell-division-cycle 2) are two names for the same protein: the first name is the newer one, but you may see either in literature (especially for genes and proteins that were named before the renaming of Cdc2 to Cdk1). Cdk1 is a mitotic Cdk found in all eukaryotes studied and is highly conserved: human Cdc2 is about 60% to 65% identical to the yeast homolog, and the human cdc2 gene can rescue yeast with a non-functional gene.
MPF
MPF (maturation-promoting factor) is a Cdk1-cyclin B complex that causes cells to exit G2 and enter mitosis.
G1-S Transition
The Restriction Point
In multicelluar animals, cells remain in G1 (or G0) until they receive a message to begin division: that is, they must be signaled to pass through the restriction point before the cell cycle begins. E2F is a transcription factor that activates transcription of genes needed for DNA replication, such as the gene for cyclin E. cyclin E combines with Cdk2 to form Cdk2-cyclin E complexes, which activate MCM helicases at replication origins, triggering the initiation of DNA synthesis.
However, in G1 (and G0), although E2F binds to its target DNA sequences, it is inactivated by being bound by the inhibitory Rb protein (The Rb gene is a tumor-suppressor gene. It gets its name from retinoblastoma, the first cancer known to be caused by mutations in the gene). The restriction point is passed when Rb dissociates from the E2F transcription factor, allowing E2F to begin activating the genes needed for DNA replication.
1. The binding of growth factors to cell-surface receptors initiates a Ras/Raf/MEK/ERK pathway that leads to the synthesis of cyclin D, and thereby formation of Cdk4, 6 / cyclin D (consisting of either Cdk4 or Cdk6 complexed with cyclin D), which is also called the G1 Cdk-cyclin complex. Cyclin D is rapidly degraded, so is present only when the cell is actively being stimulated by the growth factors.
2. The activated G1 Cdk-cyclin complex then phosphorylates the Rb protein.
3. The phosphorylated Rb protein dissociates from E2F.
4. The freed E2F transcription factor activates the genes needed for DNA synthesis, leading to the G1-S transition.
5. Later, during mitosis, Rb is dephosphorylated, and so rebinds to E2F, inactivating it.