There is nothing wrong with your staying in one of the rooms attached to the mosque or with teaching the children and using the computer, projector, blackboard and other means that help in teaching.
There is also nothing wrong with offering biscuits and juice to the children in the mosque, so long as you teach them to avoid causing any nuisance and to be keen to keep it clean.
If a person goes out of the mosque and intends to come back soon, then he does not need to pray tahiyyat al-masjid.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: I entered the mosque and prayed two rak’ahs to “greet the mosque”, then I went out intending to return. When I return, do I have to pray two rak’ahs again?
He replied: If a person goes out of the mosque with the intention of coming back, if he returns soon then he does not have to pray tahiyyat al-masjid, such as if he went out to do wudoo' and came back, or he went out to bring a book from home and came back, or he went out to speak to someone and came back. In this case he does not have to pray tahiyyat al-masjid, because the time was short. But if the time was long, then he should pray two rak’ahs. All of us, when we have prayed Zuhr, leave the mosque with the intention of coming back to pray ‘Asr, but can we say when we come back to pray ‘Asr that we should not pray tahiyyat al-masjid? No, we should pray it. What matters is that if it is for some errand and we will come back quickly, then we do not have to pray tahiyyat al-masjid and the first time we did it is sufficient. The evidence for that is that the Messenger of Allah (blessings and peace of Allah be upon him) gave permission to the one who is observing i’tikaaf to go out to his house if he needs to and then come back. This indicates that going out for a brief time is not regarded as leaving the mosque. So for the one who goes out of the mosque with the intention of returning soon, it is as if he did not go out of the mosque.
End quote from Liqa’ al-Baab al-Maftooh, 30/125
If the rooms attached to the mosque were built to be part of the mosque, i.e., the builder of the mosque intended that they should be part of the mosque in which prayers are offered, then they come under the same rulings as the mosque, so the two rak’ahs for greeting the mosque should be offered when entering them, and women who are menstruating or bleeding following childbirth should not be allowed to stay in them.
But if these room are intended to be extra rooms for teaching or holding meetings, or a place where the imam or muezzin lives, and are not intended to be a place for prayers, then in that case they do not come under the same rulings as the mosque.
If the intention of the one who built the mosque is not known, then the basic principle is that whatever is within the walls of the mosque and has a door opening into the mosque comes under the same rulings as the mosque.
This is what may be understood from the words of the scholars concerning the rooms and other facilities that are found in the mosques, taking into account the differences in the situation nowadays as compared with what was prevalent at the time of the earlier fuqaha’.
We have quoted the fatwas of the scholars concerning that in the answer to question number 118685.
There is nothing wrong with using the kitchen to prepare communal iftaars in Ramadan or to make tea or food for guests.
But with regard to making food there on an ongoing basis, that is not appropriate because it would result in filling the mosque with the smell of food, which is contrary to proper care and respect of the mosque, unless there is a wall to separate the kitchen from the mosque. This applies if the kitchen is not regarded as part of the mosque, as stated above.