What is the final accurate timing for the prayer of fajar before sunrise? for example, if the sun rises at 7:30, is it permissible to pray at 7:25?
I don't know the exact timing in your area, but Fajr is when you begin to see the whiteness of the sun across the horizon; you can pray up until the sunrises, then after that, fajr ends.
The general rule is that if one manages to complete one rak'ah prior to the adhān start of the following prayer, then one does not have to re-do that delayed prayer as it would be considered on time:
عن أبي هريرة أن النبي صلى الله عليه وسلم قال: مَنْ أَدْرَكَ رَكْعَةً مِنَ الصَّلاَةِ فَقَدْ أَدْرَكَ الصَّلاَةَ
Abu Huraira reported the Messenger of Allah (ﷺ) as saying: "He who finds a rak'ah of the prayer, he, in fact, finds the prayer."
To address your specific question about al-Subh (or al-Fajr) prayer, the Prophet ﷺ re-iterated the same rule about this prayer in particular and al-'Asr prayer, too:
عن أبي هريرة أن رسول الله صلى الله عليه وسلم قال: مَنْ أَدْرَكَ رَكْعَةً مِنَ الصُّبْحِ قَبْلَ أَنْ تَطْلُعَ الشَّمْسُ فَقَدْ أَدْرَكَ الصُّبْحَ وَمَنْ أَدْرَكَ رَكْعَةً مِنَ الْعَصْرِ قَبْلَ أَنْ تَغْرُبَ الشَّمْسُ فَقَدْ أَدْرَكَ الْعَصْرَ
Abu Huraira reported: The Messenger of Allah (ﷺ) said: "He who finds one rak'ah at dawn before the rising of the sun, he, in fact, finds the dawn prayer. and he who finds one rak'ah of the afternoon prayer before sunset, he, in fact, finds the afternoon prayer."
Having said that, delaying the prayers till the end of its permitted time is not a good act. It could easily lead to neglecting prayers altogether (see Surat Maryam 19:59). Moreover, praying on time is the best of deeds as the Prophet ﷺ said:
أخبرنا عمرو بن علي، قال حدثنا يحيى، قال حدثنا شعبة، قال أخبرني الوليد بن العيزار، قال سمعت أبا عمرو الشيباني، يقول حدثنا صاحب، هذه الدار وأشار إلى دار عبد الله قال سألت رسول الله صلى الله عليه وسلم أى العمل أحب إلى الله تعالى قال: الصَّلاَةُ عَلَى وَقْتِهَا وَبِرُّ الْوَالِدَيْنِ وَالْجِهَادُ فِي سَبِيلِ اللَّهِ عَزَّ وَجَلَّ
Al-Walid bin Al'Ayzar said: "I heard Abu 'Amr Ash-Shaibani say: 'The owner of this house - and he pointed to the house of 'Abdullah - said: I asked the Messenger of Allah (ﷺ): 'Which deed is most beloved to Allah, may He be exalted?' He said: 'Prayer offered on time, honoring one's parents, and Jihad in the cause of Allah.'"
The Prophet ﷺ always practiced what he preached at all times unless forced otherwise:
حدثنا إسحاق بن إبراهيم، أخبرنا سليمان بن حرب، حدثنا حماد بن سلمة، عن حميد، عن بكر بن عبد الله، عن عبد الله بن رباح، عن أبي قتادة، قال: كَانَ رَسُولُ اللَّهِ صلى الله عليه وسلم إِذَا كَانَ فِي سَفَرٍ فَعَرَّسَ بِلَيْلٍ اضْطَجَعَ عَلَى يَمِينِهِ وَإِذَا عَرَّسَ قُبَيْلَ الصُّبْحِ نَصَبَ ذِرَاعَهُ وَوَضَعَ رَأْسَهُ عَلَى كَفِّهِ
Abu Qatada reported that when the Messenger of Allah (ﷺ) was on a journey he got down for rest at night, and he used to lie down on his right side, and when he lay down for rest before the dawn, he used to stretch his forearm and place his head over his palm.
Scholars say that he used to stretch his forearm and place his head over his palm as this used to make his sleep restless in order to avoid oversleeping and missing the beginning time of the prayers.
Your prayer should end before 7:30 which is the time of the sunrise. 7:30 means that the sun has started to rise and you can not make Sajdah at that time as it is the time of Zawaal which ends after about 20 minutes. Makes sure that you are not in Salah at 7:30 ( or when the sun is rising).