They allow mixing in the mosque because they want people to attend

Firstly: 

Free mixing between men and women is haraam, because it leads to many negative consequences and haraam things. We have quoted the evidence for the prohibitions on mixing in the answer to question no. 1200. 

As mixing is forbidden at all times and in all places, then it is more emphatically forbidden if it is done in the mosques during the month of Ramadaan, because it goes against the shar’i aims for which mosques are established, such as maintaining and propagating Islam, calling people to good and forbidding them to do corrupt and evil things. It also goes against the reason for fasting, which is so that we might attain piety (taqwa) and avoid whims and desires.  

All the people of the mosque should denounce this evil and strive to do away with it. Those who are in charge of the mosque are especially responsible. 

No one has the right to try to justify this evil or to remain quiet about it on the grounds that not allowing mixing may cause some people to refrain from coming to the mosque. This argument is to be rejected on several fronts: 

1 – Remaining quiet and not denouncing evil when one is able to do so means that one is sinning. The Prophet (peace and blessings of Allaah be upon him) said: “Whoever among you sees an evil act, let him change it with his hand (by taking action); if he cannot, then with his tongue (by speaking out); and if he cannot, then with his heart (by at least hating it and regarding it as evil) – but that is the weakest of faith.” Narrated by Muslim, 48. 

No wise person would accept for his attendance at the mosque to be a cause of his falling into sin. 

2 – The most important role that the mosque plays is calling people to goodness and warning them against evil. Hence those who are in charge of the mosque must explain to the people that mixing is haraam, and stop them doing it. 

3 – The idea that these people will never attend the mosque is merely conjecture. Even if we assume that it does happen, the established principle among the scholars is that warding off evil takes precedence over achieving good. 

4 – A special place for women to gather can be set up, whether it is in a particular corner of the mosque or even outside in a tent or some such, and useful programs can be offered to them, which should be supervised by women. 

5 – The daa’iyah is the one who should affect how things are and strive to set things right; he should not be affected by things and then look for justification. 

Mixing is a problem that has arisen because we are far away from Islam. Efforts must be made to denounce it and put an end to it. If the first step is not taken in the houses of Allaah, then where shall we begin?!  

You can work with some of your brothers to convince the people in charge and help them to prepare a place that is suitable for women to meet, and help them to prepare suitable programs for the women. 

We ask Allaah to bless your efforts with success. 

Secondly: 

With regard to what you say about tasbeeh after each four rak’ahs of Taraweeh prayer, we have stated in the answer to question no. 50718 that this is an innovation which should not be done. 

Shaykh Ibn Baaz (may Allaah have mercy on him) was asked: What is the ruling on raising our voices in sending blessings upon the Prophet (peace and blessings of Allaah be upon him) and praying that Allaah be pleased with the Rightly-Guided Caliphs between the rak’ahs of Taraweeh? 

He replied: 

There is no basis for that – as far as we know – in sharee’ah, rather it is an innovation and should not be done. The affairs of the later generations of this ummah will never corrected except by that by which the affairs of the earlier generations were corrected, which is following the Qur’aan and Sunnah and the way of the earliest generations of this ummah (the salaf), and avoiding whatever goes against that. End quote. 

Majmoo’ Fataawa Ibn Baaz, 11/369. 

And Allaah knows best.