All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
If you had rented this place in order to perform the prayers in it, then you should be keen on performing the five obligatory prayers in it so that the purpose for which you had rented it will be fulfilled. We also recommend that you take your children with you to pray so that they would get attached to praying and to the Muslim congregation. This would most likely render them righteous in such societies that are full of evil.
As for your question whether or not this rented place is considered a mosque and whether all the prayers should be performed in it, then the answer is no. The rented Musallas (prayer rooms, prayer places) and other places where the prayer is performed, like those in the workplace; if a person prays in them with the congregation, he will get the reward of the congregational prayer, but they are not considered a mosque and they do not take the same ruling as the mosque.
The difference between a Musalla and a mosque is that a mosque is set up as a Waqf (endowment) for this purpose eternally, whereas a Musalla is not set up as a Waqf.
Additionally, they do not take the same ruling of a mosque in terms of the prohibition of a menstruating woman and a person in a state of Janaabah (major ritual impurity) to stay in them, and praying the two Rak'ahs upon entering them is not the same as that of the mosque. Shaykh Ibn ‘Uthaymeen and other scholars stated this.
However, even though this Musalla is not a mosque, a Muslim who is keen on preserving his religion should be keen on performing the prayers in congregation wherever it is possible for him to do so, whether in these Musallas, or at home, or at his work place.
Many jurists who are of the view that the congregational prayer is an obligation did not restrict this to the mosque; rather, they stated that it can be performed at the mosque or elsewhere.
Allah knows best.