Is it permissible to violate residency conditions in Muslim countries (such as not working or staying for a short period) or to purchase sponsorship for entry, if the intention is to emigrate for the purpose of seeking knowledge and fatwa, with the assurance of safety from harm and not causing harm to others?

All praise is due to Allah.

Migration (Hijra) is not obligatory for a Muslim residing in a non-Islamic country if they are able to openly practice their religion and observe its rituals. Migration from a land of war is divided into three categories: obligatory for the capable person who cannot openly practice their religion; not obligatory for the incapable person (such as one who is oppressed); and desirable for the capable person who can openly practice their religion (to enable them to participate in jihad and increase the number of Muslims).

Residing in an Islamic country is better for a Muslim and their family, unless one remains for the purpose of da'wah (inviting to Islam) and spreading knowledge. Indeed, the Prophet, peace and blessings be upon him, permitted some Companions to remain in their homelands to teach their people. He also said to a Bedouin who asked him about Hijra: "Work from beyond the seas, for Allah will not diminish anything from your deeds," meaning that the reward for his deeds will not be lessened, no matter where he is.

If moving to an Islamic country is not possible except by violating conditions, then the advice is to remain in the current country until the Muslim finds a way out, unless they are compelled to migrate. Participating in insurance in a non-Islamic country—if it is a condition for residency—is permissible when necessary.