How do scientists justify facts on galaxy A1689-zD1 and put say it age?

There is a mathematical relationship between the red shift of the spectrum of a galaxy and it's age. That relationship is an estimate based on the best available information. The relationship is one that has been carefully worked out over the past 80 or so years based on fairly well known but shorter distances. It has been subjected to some corrections. The public media might say 13 billion light years or whatever, the astronomers probably say something like 13.2 billion plus or minus 50,000,000 light years.

It is also time you realised what "theory" actually means. It means a set of ideas that attempt to explain some observation or phenomena, which is backed up by EVIDENCE. That is what a scientific theory is. You are confusing "theory" with speculation, guess, conjecture or hypothesis. They are not the same.