What is the ruling on applying for residency in a country that does not rule by Allah's law, especially if it involves conditions derived from man-made laws, and is this considered an declaration of loyalty to Taghut?

There is no place for such a question in the reality of most parts of the world where the rulings of Sharia (definition) are not applied. A Muslim should not approve or be content with falsehood; rather, he should detest and denounce it with his heart, and by doing so, he would have fulfilled his duty. After that, he would not be described as aligning with or being loyal to tyranny, because loyalty means love, support, approval, and following. The Prophet, peace and blessings be upon him, narrated: "If a sin is committed on earth, he who witnesses it and dislikes it is like one who was absent from it, and he who was absent from it and approved of it is like one who witnessed it." He also said: "Indeed, rulers will be appointed over you, and you will recognize some of their deeds and denounce others. So, whoever dislikes it has absolved himself, and whoever denounces it has saved himself, but whoever approves and follows." This applies to one who can practice his religion and is safe from trials, so how about one who does not have this capacity? "Allah does not burden a soul beyond its capacity."