Women

Yes it is allowed for woman to take a bath at night, without any difference with man. It might be a cultural reason in some places behind this question.

Wassalam.

Lowering the gaze has been emphasized upon in Quran very explicitly as we read the order of Allah to the Prophet (Tell the believing men to lower their gaze) Sura Al-Noor, verse 30. Same order was given for the believing women to lower their gaze (Sura Al-Noor, verse 31).

Those who remember Allah and obey His orders as obligatory order, Lowe their gaze and protect themselves from looking at sinful scenes.

We have many authentic Hadeeths in this regard which also emphasize on lowering the gaze and the punishment of those do not lower their gaze.

Wassalam.

Women are not only allowed but also encouraged to seek knowledge with out any limitation. Women can reach to highest degrees of knowledge which is known as Ejtihad إجتهاد.We have in our Shia history many women reached to the high degree of Ejtihad. When the women becomes highly learned in Islamic sciences up to Ejtihad, she will not be dozing Taqleed for other Mujtahids but she as a Mujtahidah has to act upon her own verdicts.

The post of Marja' of Taqleed has big burden which is not for any woman to bear with her delicate nature as a female and her physical abilities and circumstances e.g. menstruation cycle, pregnancy, breast feeding etc.

Woman can reach to the highest degrees of Islamic knowledge with out making her facing challenges which are beyond her abilities and comfort.

Islam is not a male dominated religion, as Islam teaches that the human beings are created in male and female (Sura 49, Verse 13). Islam granted the woman since more than 1400 years full rights which were never given to her by any other religion or ideology. Europe till recently, treated with women as incomplete human beings and used to give her less than the salary of similar job man. Women were not allowed to study nor to do business or even own. 
Islam put pious women in the highest rank above millions of men. Lady Fatimah (AS) is the best human being after her father Muhammad (SAWA) and her husband Ali (AS).

Wassalam.

No Marja and no pious scholar will allow any non Mahram female to kiss or touch his hand directly. It is not allowed and the Marja will never allow it.

Wassalam.

All men and women in Paradise will wear best dress as it is mentioned many verses in Quran as Sondos and Estabraq (Sura 18, verse 31 and Sura 44, verse 53) . Sondos and Estabraq are types of best pure silk.

Woman in Paradise will wear the best clothes with best dignity and honour of full Hijab. No sinful thought or act can be imagined in Paradise because they will be purified from all wrong and evil thoughts.(Sura 7, verse 43 and Sura 15, verse 47).

Wassalam.

Muslim man must lower his gaze always and never look at females who are not wearing Hijab. Sitting with non-Hijabi women should not encourage them or give wrong message endorsing their act in neglecting the orders of Allah in Hijab.

Wassalam.

Muslim men have been ordered in Quran to lower their gaze and not look at Nin Mahram women who are not covered by proper Hijab. (Tell the believing men to lower their gaze and protect their private parts (from sinful sexual acts). That is purer for them. (Sura Noor, verse 30).

Next verse says: And tell the believing women to lower their gaze ( from looking at non Mahram men) and protect their private parts.(from sinful sexual acts).(Sura Noor, verse 31).

These orders aim preventing acts which can initiate or cause sinful sexual acts and maintaining the purity and piousness in the hearts of men and women when dealing with each other. It is a type of prevention which is better than cure. Sinful looking initiates sinful desire which can be a step towards sinful sexual act, that is why, lowering gaze has been clearly ordered in Quran and authentic Hadeeths.

Wassalam.

It is not allowed to take part in a trip or gathering where men and women are together with out full Hijab.

Wassalam.

Yes. Woman can be a martyr in different ways. Mother of Ammar ibn Yaasir was martyred by the disbelievers in Makkah and mother Wahab was martyred in Karbala and any Momina who dies with the love of Ahlul Bayt (AS) will die as a martyr. (Tafseer Al-Zamakhshari , v. 4,  Tafseer Al-Raazi) v. 27.

Wassalam.

Human dignity of woman does not allow her to be with more than man at the same time. It is not only in Islam, but in all main religions that woman is allowed to marry one man at the same time. More than one man at the same time means destroying the dignity of the woman and shaking her self respect. Obviously, if a woman is allowed to marry more than one man at the same time, the society will lose trust in the belonging of the children to their father as no one will know who is the real father of the child. That itself is a big damage to the human sonciety.

Wassalam.