So the sperm finds an egg, fertilization occurs and a zygote is formed. Yes while it is still in the archegonium and at the tip of the female gametophyte a new plant, the sporophyte, starts to grow. Yes a mature female moss plant consists of a diploid sporophyte growing out of a haploid gametophyte. Yes since this is where meiosis occurs, four cells are produced. The cluster of four is often called a tetrad.
Spores come from a sporophyte, but they grow into something else.
Yes a spore germinates to form a filamentous protonema that does not survive. In a sense, although the plants that we recognize as gametophytes grow out of something else.