Hajj

The main reason is that it was not mandated by the Prophet; in fact, he prayed with his wife and daughter there.

Also, (a) because to do tawaf you have to go all the way around the Ka'bah, and that is difficult to do with segregation, similarly it might be challenging to arrange gender segregation for sa'i between Safa and Marwah, and (b) perhaps it could cause hardship for people travelling together. 

While it is true that there are some people with sick minds who misuse the arrangement there, by and large, I think most Muslims behave themselves properly there and respect the sanctity of the place, and the rules of ihraam (forbidding certain things between genders) also provide extra incentive to behave correctly, even if they may run into each other by accident.

That being said, wearing an extra outer garment can help provide more personal space without inconveniencing others.
 

A cousin is not mahram unless you are married to them or is a secondary reason why you are mahram (such as being milk siblings). However you do not need a mahram to go to hajj.  This is just a legal requirement for the visa from Saudi and is not required in Shi'i fiqh.

I hope you are able to perform the hajj successfully, God willing!

Hajj is prescribed for both men and women in Islam and is obligatory one time on anyone who is able to perform it.

If she is not able to perform hajj (because she cannot afford it, because it would cause hardship, because she is physically unable to, because she cannot get a visa, or for other reasons), then it is not obligatory right now. 

Sometimes we are not able to do things in life because of our circumstances, and this is part of life. We are not responsible for the things we cannot do. 

As the Qur'an says, Allah does not intend to burden us!

If she is not able to go, it is good for her to pray for Allah to grant her the opportunity to go in the future and to have the intention to go if it becomes possible. How often are prayers to go to hajj answered!

If her circumstances change so that she becomes able to do it, then it becomes obligatory.  

However the intent behind Islamic law is that everyone who can perform hajj does so regardless of whether they are a homemaker or not. 

Yes, this is in the book Kamil al-Ziyarat:

The narration is as follows: I said to Imam Sadiq (A), “May I be sacrificed on you, I cannot afford to go to Hajj every year.” The Imam (A) replied: If you ever wanted to go to the Hajj but could not afford it, go to the Ziyarat of Imam Husain (A), for indeed it will be recorded as a Hajj for you. And if you ever wanted to go for Umrah but could not afford it, go to the Ziyarat of Imam Husain (A), for indeed it will be recorded as an Umrah for you.

As for why? Allah knows best.

However, some thoughts. First, it is not neglecting hajj, only discussing the situation where it is impossible.

Second, it is also said that worshipping at Masjid al-Kufa has a similar reward to hajj, and Masjid al-Kufa is associated with Imam Ali (A) so this general idea is not only said about Karbala (although it is more emphasized for Karbala).

Third, if someone from afar cannot perform hajj, then probably they also cannot go to Medina and visit the Prophet (S), so it is not worth mentioning here.

Fourth, there are many narrations that indicate there is something spiritually unique about Karbala and where Imam Husain (A) was laid to rest, and visiting Imam Husain (A) is emphasized above visiting the resting places of the other Imams. So, it is good to trust in that and assume there is in fact a hidden merit to visiting Karbala which is not found in other places. There are many aspects of the spiritual world that we don't see with our physical eyes and in our everyday reality. Of course, it is also very good to visit Imam Ali (A)!

The best way to answer this question for one's self is to go to all these places (hajj, Najaf, Karbala) and develop a personal view. If you have not, may Allah bless you with the opportunity!