How are dominant and recessive traits being passed on to offspring?

Answer 1

They're passes on with the DNA each time an organisms reproduces. In bacteria, the "offspring" might get an exact copy of the parent's DNA since they reproduce by fission (splitting in two), but even in bacteria, sometimes they "exchange" pieces of DNA. This also happens in plants that "divide" by reproducing using stolons, rhizomes, bulbs with offshoots, etc.

In plants and animals with sexual reproduction (involving meiosis), the offspring get one chromosome from each parent when the gametes join (fertilization). Dominant traits are the ones that are expressed if they're present, but it's possible that an organisms can get both one dominant and one recessive gene (depends on the genotypes of the parents), but the recessive gene won't be expressed if one for the dominant trait is present - both genes must be for the recessive trait for it to be expressed.