For mitosis What happens, in G1 phase the DNA content is 2C (for diploid organism). The DNA content indeed gets 4C after S phase that is why you get 2 chromatids per chromosome before next metaphase. So after S phase is completed, the DNA content is actually 4C although the chromosome number is said to be 2n. after metaphase the chromosomes are split and then daughter cells get one of each sister chromatid, then each daughter cell gets DNA content of 2C.
Genotypes in diploid/haploid cells under mitotic/meotic cell divisions
here is a diagram from there (highly diagrammatic) about what happens with chromosome numbers during cell cycle
"By chromosomes in the S phase, do we mean actual "rod-shaped"
structures that are formed from chromatin condensation or it's just
the chromatin threads taking the shape of chromosomes?"
it is just the Chromatin threads (DNA + some proteins) loosely spread in nucleoplasm (karyolymph) in an entangled, noodle like manner.
Yes, there is a visibility issue, during interphase (G1 S, G2) the euchromatin regions of the chromatin fibres stays in a decondensed form of 30 nm or somewhat more condensed 60 or 130 nm (The Cell, G.M. Cooper, ed-4) whereas maximum resolution (d) achieved by light microscope is somewhere about 200 nm
two chromatids per chromosomes
Each chromatid stands for 1 linear DNA molecule; regardless of condensed or decondensed. It is mostly decondensed in interphase, and mostly condensed in M phase (P-M-A-T).
Here is an image of 2 chromatids in 1 chromosome :
This image from shutterstock well explained the configuration of chromosome just after telophase (1 chromatid per chromosome) and after replication to metaphase (2 chromatids per chromosome)
In the following sketch; the changes through chromosome number and form is shown diagrammatically. The terms A and B here are just for labelling purpose, they aren't any technical terms.
image A (1-5) shows what happens wth DNA; and
Image B (1-5) shows somewhat realistic appearance of chromosomes (and chromatids);
during cell cycle.
see also: Interpretation of picture of human chromosomes