Is the chromosome number of zygote, embryonal cells and adult of a particular organism always constant? How is the constancy maintained in these three stages?

Yes, the chromosome number of zygote, embryonal cells and adult of a particular organism is always constant.

The gametes have half number of chromosomes as compared to that normal body cells. Reduction division (meiosis) takes place during gamete formation which halves the number of chromosomes in both male and female gametes. The original chromosome number (as in parent) is restored after fertilisation in sexual reproduction. This process maintains the chromosome number and the amount of DNA in each generation.