Is it permissible for me to use "wasta" (connections/mediation) to transfer to another city given the difficult housing and financial circumstances, and does the ruling differ if the transfer is not to my original city, especially since there is a colleague who wants to transfer but does not have "wasta"?

The summary of the answer is encapsulated in several points: First: Using intercession to attain a right that cannot be obtained except through it is permissible, as Allah says: {Whoever intercedes with a good intercession will have a share of it}. Second: If everyone has an equal right that cannot accommodate them all, then fairness dictates drawing lots to distinguish the deserving when those entitled are equal. Third: If there is no lottery and one of the entitled fears losing their right, they may use intercession to obtain it, even if it leads to depriving others, because they are taking their right, not the right of others. Regarding the saying "equality in injustice is justice": please refer to Fatwa number: 69252.