House

You should not invite them to your house as Islamic law strongly discourages or condemns the situation where a woman and a non-mahram male are alone together. (Whether you would be alone in the house or alone in a room.) There is no need for them to be at your house. This will prevent any sort of problems.

Also remember that your intentions are only your own, and you don't have any guarantee about what the other person is intending or experiencing. 

This subject has been broached here, and perhaps the fact that you are asking about it will encourage others to add some thoughts on it.

https://www.al-islam.org/ask/why-does-a-husband-have-the-power-in-islam-...

The person who is asking this may be familiar with the jurisprudential model. That is, insofar as it is the husband's right to have physical intimacy, he can prevent the wife from leaving the home (because this might interfere with his right). And his right is due to being the breadwinner and the wife being a financial dependent.

This is obviously a theoretical or idealized model of a marriage and doesn't take into account the varieties of how people live as well as the nuances of real life (for instance, that, oftentimes, the man isn't the sole breadwinner, and/or that the household labour that woman often do is equally important to the survival of the household or children; and, generally, both men and women have an interest in physical intimacy).

In practice, I would say that most men do not imprison their wives and don't do this literally. But, occasionally, a man does do that in the name of Islam, thereby depriving the wife of opportunities for seeing family or friends, education, attending religious gatherings, and knowing what is happening in society. And, in the context of the religious community, it is very difficult for anyone to object; a woman fears that if she disobeys him and goes outside, she will go to hell. (I had a friend in the US who was kept inside by her husband for decades; after he passed away, she didn't even know what an ATM/cash machine was). And this can be especially harmful if a girl marries young. Of course, in some places, this is also done because of lack of security in society and fear for the wife's safety, and this should also be acknowledged (that is, it isn't always due to overdominance).

Furthermore, the fact that a wife has to ask for permission to leave could be seen as demeaning to her dignity. I am sure most men would not tolerate it if someone told them they could only move about with a woman's permission!

As you might glean from my response, I am not in favour of this paradigm, and I am more in favour of the late Sayed Fadlullah's view that marriage should not be imprisonment, and that this particular ruling should be reconsidered. Sometimes, we just pass on things from the classical era without questioning them (such as the ruling of purity of Ahl al-Kitab, which was questioned in the modern era) until the time comes when we realize we do need to give it another look. I don't think the Prophet intended to disadvantage women, especially since so much of his message was about social reform and he had a special concern for improving the situation of women. And when we look at hadith from the time of the Prophet (S), we see that women were quite involved in what was happening around them and in the community of the Prophet (S), rather than being solely at home and uninvolved in anything around them. 

However, I do acknowledge that my own view is non-mainstream (that is, it is more of what is considered today a reformist view) and that the mainstream view is that this is because Islam, as a perfect system, provides an ideal model of marriage, and if people follow it they will have harmony. And that this authority given to the man provides order and structure for society and prevents moral corruption and so on. 
 

It is not forbidden, although, customarily, some people might consider it better to wait.

Hadith focus on mourning on the day of Ashura and refraining from worldly affairs on Ashura, or for the first 10 days of Muharram, rather than the entire month of Muharram. I would personally advise against buying a house on the actual day of Ashura, although again it is not forbidden. 

Anyway, you have to live somewhere, and if you need somewhere to live now, or if your best option to buy the house now, you have to make the choice that is best for you. You are better able to serve Imam Husain if you are not homeless!