Praise be to Allah.
One who is unable to stand in the obligatory prayer may pray sitting down, and one who is unable to bow and prostrate may lean forward for these movements, making the prostration lower than bowing.
The one who prays sitting may sit on the ground or on a chair, according to whatever is easier for him. He is called, like others, to strive and compete to be in the first row, and it is not permissible for anyone to prevent him from doing that, just as it is not permissible for him to be on his own behind the row if there is space in the row.
What you have mentioned about the custom of putting the chairs at the back of the mosque is a bad custom, as it deprives the one who sits on a chair of the reward of being in the front rows, and it is contrary to the Sunnah, which is to complete the first row then the next row, and for the rows to be close together.
Al-Bukhaari (615) and Muslim (437) narrated from Abu Hurayrah that the Messenger of Allah (blessings and peace of Allah be upon him) said: “If the people knew what there is (of reward) in the call and the first row, and they could find no other way then drawing lots, then they would draw lots. If they know what there is (of reward) in coming early to prayer, they would compete for it. If they knew what there is (of reward) in ‘Isha’ and Fajr prayer, they would come to them even if they had to crawl.”
An-Nasaa’i (811) and Abu Dawood (664) narrated that al-Baraa’ ibn ‘Aazib said: The Messenger of Allah (blessings and peace of Allah be upon him) would enter between the rows from the back to the front, and he would touch our chests and shoulders and say: “Do not differ, lest your hearts also differ.” And he used to say: “Allah and His angels send blessings on the front rows.” Classed as saheeh by al-Albaani in Saheeh an-Nasaa’i.
Ibn Maajah (997) narrated it as follows: “Allah and the angels send blessings upon the first row.”
Ahmad (12352), Abu Dawood (671) and an-Nasaa’i (818) narrated from Anas that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Complete the first row, then the next, and if there is any deficiency, let it be in the last row.”
The wording of Abu Dawood is: “Complete the front row, then the next, and if there is any deficiency, let it be in the last row.”
Classed as saheeh by al-Albaani in Saheeh Abi Dawood.
An-Nawawi (may Allah have mercy on him) said: Our companions and others are agreed that it is mustahabb (recommended) and encouraged to be in the first row, and there are many hadiths that speak of that in as-Saheeh. They also agreed that it is mustahabb to stand to the right of the imam, to close any gaps in the rows, and to complete the first row, then the next, until the last row; a new row should not be started until the one in front of it is completed.
End quote from al-Majmoo‘ (4/301).
Abu Dawood (667) and an-Nasaa’i (815) narrated from Anas (may Allah be pleased with him) that the Prophet of Allah (blessings and peace of Allah be upon him) said: “Line yourselves up tightly, and make the rows close together, and put your necks in line.” Classed as saheeh by al-Albaani in Saheeh Abi Dawood.
As-Sindi (may Allah have mercy on him) said: “Line yourselves up tightly” means stand close to one another in a straight line; “and make the rows close together” means keep the distance between rows small, so that the rows are close to one another. End quote.
Al-Mannaawi (may Allah have mercy on him) said: “and make the rows close together” means: in such a way that is no room for another row between two rows, so that the Shaytaan will not be able to pass in front of you.
End quote from Fayd al-Qadeer (4/7).
If putting a chair will lead to creating a gap in the row, then it should be placed at the end of the row, on the right or the left, or in the middle of the row whilst striving to close any gaps.
Ahmad (5724) and Abu Dawood (666) narrated from ‘Abdullah ibn ‘Umar that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Straighten the rows, for you are standing in rows like the angels; put your shoulders in line, fill the gaps, and do not leave gaps for the Shaytaan. Whoever connects (completes) a row, Allah will connect him to His Mercy.” This hadith was classed as saheeh by al-Albaani in Saheeh Abi Dawood.
If it is possible to put the chair in a place in the row that will not affect those who are standing behind it, then that is best. The worshippers should cooperate in that regard, in a kind and gentle manner, with a good attitude and good manners. The person who is unwell should not object to that, for among the words that have been narrated from the Prophet (blessings and peace of Allah be upon him), in which he used to give instructions to straighten the rows for prayer, are the words: “Be gentle with the hands of your brothers.” Narrated by Ahmad and Abu Dawood; classed as saheeh by al-Albaani in Saheeh al-Jaami‘ (2721).
That is, if someone is standing in the row, and someone who wants to straighten the lines gestures to him with his hand to make himself in line with the row, or places his hand on his shoulder, then let him move accordingly and put himself in line with others.
If someone wants to join a row, then they should make room for him.
End quote from Nayl al-Awtaar (3/231),
In Mirqaat al-Mafaateeh (3/853) it says: What is meant is: If someone is standing in the row, and someone tells him to straighten the row, or puts his hand on his shoulder, then he should yield and not be arrogant. End quote.
If that is not possible, then there is a way to avoid interrupting the row or causing disruption in the row behind the chair, by moving the chair a little forward, so that the back of the chair will be in line with the row of worshippers, as we often see.
This is emphasized further if the one who is sitting on the chair will be sitting on it from the beginning of his prayer, or for most of his prayer. In that case placing the chair in this manner is what is required, so that his shoulders and body will be in line with the row when he is sitting on the chair. This is what is required of him, and it does not matter that his knees or feet will be ahead of the row.
Placement of the chair in this manner will complete the row in which he is, and will not cause any annoyance to others or disruption in the row behind.
What appears to be the case is that the chair should be placed like that, with the back of the chair in line with the row in which it is placed, even if he is going to stand during part of his prayer and sit on the chair during part of it, so that he will not need to move the chair during the prayer, with what that will cause of distraction and disturbance to others.
If he is going to be in front of the row when standing, he is excused for that, and he will be forgiven for standing in front of the row for part of his prayer. Placing the chair like this is better than placing it further back from the beginning of the prayer, because in that case he will also not be in line with [the rest of the row] throughout the prayer, and will also cause disruption in the row behind.
Whatever the case, what all worshippers should do is cooperate in righteousness and piety, which includes cooperating in making the rows complete.
That also includes cooperating to resist this bad habit that has become widespread in some countries, which is placing the chairs or fixed seating at the back of the mosque, where those who need to sit during the prayer pray, because it is contrary to the Islamic etiquette of completing the rows from front to back. This habit has a negative effect and there are many things wrong with it, as may be seen by looking and observing those who sit there at the back of the mosque.
For more information on the rulings on praying on a chair, please see the answer to question no. 50684.
And Allah knows best.