When do 3 germ layers form in embryo? And why is there only one mature egg formed?
I'll bifurcate the question and answer each part one at a time.

Germ layer formation: Gastrulation is a phase early in the embryonic development of most animals, during which the single-layered blastula is reorganized into a trilaminar ("three-layered") structure known as the gastrula. These three germ layers are known as the ectoderm, mesoderm, and endoderm i.e. the layers are formed with gastrula, or simply gastrula is the structure which contains these 3 layers.1



Polar body egg: Polar bodies generally don't have the ability to get fertilized as they are just residual parts of oogenesis. They are formed just to maintain haploid number in oocyte, so they don't even get sufficient amount of cytoplasm to survive (most of the cytoplasm is given to oocyte to provide nourishment to embryo). So they apoptose and disappear, but in some cases (like this), they are considered important and can even lead to unusual phenomena like polar body twinning, in which twinning occurs when the egg cell and polar body are both fertilized by separate sperms.2.

Cross over: In oogenesis (or specifically, ootidogenesis), cross over occurs during the conversion of primary oocyte to secondary oocyte and first polar body. Meiosis I of ootidogenesis begins during embryonic development, but halts in the diplotene stage of prophase I until puberty. For those primary oocytes that continue to develop in each menstrual cycle, during puberty, synapsis occurs and tetrads form, enabling chromosomal crossover to occur. As a result of meiosis I, the primary oocyte develops into the secondary oocyte and the first polar body.3

(In mammals, the first polar body normally disintegrates before dividing, so only two polar bodies are produced.)

References:


https://en.wikipedia.org/wiki/Gastrulation
https://en.wikipedia.org/wiki/Polar_body
https://en.wikipedia.org/wiki/Oogenesis
Polar bodies—more a lack of understanding than a lack of respect
Rare Forms of Twinning