Conditions of women going out to the mosque
Question

Is it permissible for women to go and pray tahajjud in the mosque without a mahram when the mosque is next to the house and the men in the family do not do this prayer?.
Praise be to Allah.It is permissible for women to go to the mosque and pray, subject to certain conditions. It is not one of these conditions that she should be accompanied by a mahram, so there is nothing wrong with her going to the mosque to pray without a mahram. 
It says in Fatawa al-Lajnah al-Daa’imah, 7/332: 
It is permissible for a Muslim woman to pray in the mosque and her husband does not have the right to stop her if she asks him for permission to do that, so long as she is properly covered and no part of her body is showing that it is forbidden for “strangers” (non mahrams) to see. It was narrated that Ibn ‘Umar said: I heard the Messenger of Allah (peace and blessings of Allah be upon him) say: “When your womenfolk ask you for permission to go to the mosque, give them permission.” According to another version, “Do not forbid women their share of the mosques if they ask you for permission.” Bilaal – a son of ‘Abd-Allah ibn ‘Umar – said, “By Allah, we will stop them.” ‘Abd-Allah said to him, “I say ‘The Messenger of Allah (peace and blessings of Allah be upon him) said…’ and you say, ‘We will stop them’?!” Both reports were narrated by Muslim. 
If the woman is uncovered and any part of her body is showing that it is forbidden for “strangers” (non mahrams) to see, or she is wearing perfume, then it is not permissible for her to go out of her house in this state, let alone go out to the mosque and pray there, because of the fitnah (temptation) involved. Allah says (interpretation of the meaning): 
“And tell the believing women to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts) and not to show off their adornment except only that which is apparent (like both eyes for necessity to see the way, or outer palms of hands or one eye or dress like veil, gloves, headcover, apron), and to draw their veils all over Juyoobihinna (i.e. their bodies, faces, necks and bosoms) and not to reveal their adornment except to their husbands…” [al-Noor 24:31] 
“O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks (veils) all over their bodies (i.e. screen themselves completely except the eyes or one eye to see the way). That will be better, that they should be known (as free respectable women) so as not to be annoyed. And Allah is Ever Oft-Forgiving, Most Merciful” [al-Ahzaab 33:59]
Zaynab al-Thaqafiyyah used to narrate that the Messenger of Allah (peace and blessings of Allah be upon him) said: “If any one of you (women) attends ‘Isha’ prayer, let her not put on perfume that night.” According to another report, “If any one of you (women) attends the mosque, let her not put on perfume that night.” Narrated by Muslim in his Saheeh. 
It was proven in saheeh ahaadeeth that the women of the Sahaabah used to attend Fajr prayer in congregation, covering their faces, so that no one would recognize them. It was proven that ‘Amrah bint ‘Abd al-Rahmaan said: I heard ‘Aa’ishah, the wife of the Prophet (peace and blessings of Allah be upon him), say: “If the Messenger of Allah (peace and blessings of Allah be upon him) had seen the way the women are behaving, he would have forbidden them to go to the mosque as the women of the Children of Israel were forbidden.” It was said to ‘Amrah: Were the women of the Children of Israel forbidden to go to the mosque? She said: Yes. Narrated by Muslim in his Saheeh. 
These texts clearly indicate that if the Muslim woman adheres to proper Islamic etiquette in her dress and avoids adorning herself in ways that will provoke fitnah and affect those of weak faith, there is no reason why she should not pray in the mosque. If she appears in such a way that evil people and those in whose heart is a disease will be tempted by her, then she is not allowed to enter the mosque, rather she is not allowed to leave her home and attend the mosque. 
Shaykh Ibn ‘Uthaymeen said in Majmoo’ al-Fataawa, 14/211: 
There is nothing wrong with women attending taraaweeh prayers if there is no danger of fitnah, subject to the condition that they go out looking decent and not wearing adornments, make-up or perfume. 
In his book Hiraasat al-Fadeelah (p. 86), Shaykh Bakr Abu Zayd listed the conditions for women to go out to the mosque. He said: 
Women are allowed to go out to the mosque according to the following rulings: 
1-That there is no risk of them tempting others or being tempted
2-That their attendance will not lead to anything that is forbidden according to sharee’ah
3-That they do not jostle with men in the street or in the mosque
4-That they should go out not wearing perfume
5-That they should go out wearing hijab, not making a wanton display of their adornment
6-A door should be set aside in the mosque just for women to enter and exit, as mentioned in the hadeeth narrated in Sunan Abi Dawood and elsewhere.
7-The women’s rows should be behind the men
8-The best rows for women are those at the back, unlike the case for men
9-If the imam makes any mistake in his prayer, men should say “Subhan Allah!” and women should clap
10-The women should leave the mosque before the men, and the men should wait until the women have dispersed to their homes, as mentioned in the hadeeth of Umm Salamah (may Allah be pleased with her) in Saheeh al-Bukhaari and elsewhere.