You do not need to pay khums under these conditions.
The itinerant distance is the criterion for the calculation of distance in religious issues. The reason for this ruling is that what could generally be understood from traditional proofs (ḥadīths) is the itinerant distance which is calculated based on the normal way traversed by people. This is particularly emphasized because usually the direct way between two locations is shorter than the normal itinerant distance between them that people usually wend. Therefore if the direct way were the criterion for calculating distance in religious issues then it should have been explicitly mentioned in narrations.
Under the mentioned condition, which is a necessary condition, there is no issue with this kind of purification.
If the same illness led to his death and he was not able to perform the qaḍā of his prayers and fasts then there is no need to perform their qaḍā. However, if he regained his health and was able to perform them but he did not, it is obligatory on his elder son to make up for them. But those that he had performed incorrectly although it is not obligatory for his elder son but he had better perform their qaḍā.
Performing these acts in the mosque is not prohibited and there is no need for the permission of the religious authority. However, attending to worldly acts in the mosque is considered to be makrūh and if these acts disturb the worshippers in the mosque they are prohibited.
About mosques that are situated in the middle of streets, there is no way that they can be demolished, both because they are mosques and because the property on which they have been built has been devoted by someone to the mosque. Thus anyone who demolishes such a mosque should pay the price for that by building another mosque or repairing other mosques. In fact, this is an example of wasting the property itself and until there is no necessity, destroying a mosque is strictly impermissible. As regards desolate mosques in deserts and desolate villages, they should be kept in a way that they are not desecrated.
The obligatory precaution demands that infidels be prevented from entering mosques except when they intend to research about Islam and like that.
Precaution demands that the permission of the custodian be obtained but if the custodian of the mosque disagrees with what is in the interest of the mosque or he doesn’t pay attention to them, the board of trustees can obtain the permission of the religious authority in that matter.
Demolishing a mosque is not permissible but if it collapses itself it is permissible to use its materials to rebuild the mosque of that village or other places.
This is not permissible, but it is permissible to sell them and use the money for other needs of the mosque.
It is sufficient.
It is not permitted, but if the person who devoted the land for the mosque has allowed it, it will be permissible. However, there is no issue with constructing things like libraries in the basement of the mosques.