Prayer timetable says that dawn is at 3:53am, and the sun rises at 5:45am. The Adhan for Fajr is very close to 5:45, something like 20minutes before. My question is, if I am going to fast(Dhul Hijja) I can eat until dawn, can I pray Fajr before the Adhan around 4? Why is Adhan so late, or is it the timetable that is making the problem? If it is, if dawn is later than it is written, why does it say that it is that early and people can't eat anymore from that early time?
Without more information we can't really tell you whether the timetable you are consulting is correct. The timings vary depending on your location and the time of the year. You can consider consulting multiple apps and websites to make sure that the timing is correct.
Assuming it is correct, you must start the fast at dawn. You can pray Fajr at anytime between dawn and sunrise, and it is not required for the azan to have been called beforehand. For reference the prayer times are based on the hadith of Gabriel:
وصلى بي الفجر حين حرم الطعام والشراب على الصائم
he prayed the dawn prayer with me when food and drink become forbidden to one who is keeping the fast
@Uma has pointed at the fact that prayer times depend on location and the time of the year and that we can't answer your question without more information. Here brief comments/answers to your partial questions and maybe some additions:
If I am going to fast(Dhul Hijja) I can eat until dawn, can I pray Fajr before the Adhan around 4?
If in the mosque the adhan is held after the time of fajr you may be allowed to pray fajr once you are certain that the time has entered be it 4 a.m. or 4:15 a.m. or even 3:55 a.m. uncertainty in this case leads to invalidity of a prayer.
Why is Adhan so late, or is it the timetable that is making the problem?
You should ask the imam of your mosque. Many things in Islam have clear rulings and in some cases their are options for example it is allowed to call for fajr even before the time to let people wake up and prepare themselves, if the community has been informed. Otherwise it is not allowed to call for a prayer before the actual time has entered. So only the people of the mosque and its community can tell you what actually is the case there in.
In many mosques in Europe they decide to pray fajr later in the summer to leave people time for rest and so that working people may pray and go to work.
If the time table was distributed at the mosque one could assume that it is with the agreement of the imam/community, but it is safer to ask.
If it is, if dawn is later than it is written, why does it say that it is that early and people can't eat anymore from that early time?
Fast is a worship we do for Allah and nobody would like to risk doing it badly or starting it badly, that's why many people keep a safe time zone between adhan and the time tables (as time tables may have an inaccuracy of a few minutes +/-)
You may also refer to the chapter on the prayer times in al-Muwatta' for more information.
Note that if you have certainty that the time for fajr has entered you are neither allowed to eat nor drink if you want to fast that day.