To take your questions in reverse order, the process did change over time. In the early period, which ended roughly in the 16th century, the Ottomans practiced what has been referred to as "open succession." Basically, upon the Sultan's death, any male offspring of the reigning Sultan would vie militarily for the throne, killing their brothers and half-brothers until only one remained. The one who remained would become the next Sultan. Because of this practice, the son stationed closest to the capital had an advantage, as information about the Sultan's death would reach that son first, and they were able to race to the capital and marshal forces before any other claimants. Thus, the closest son to the capital was often considered to be the reigning Sultan's favorite, and being stationed in, say, Nicaea might provide a significant advantage in getting to Constantinople before the other claimants.

Later, the claimants would even begin to struggle for succession prior to the death of the Sultan, often backed by their respective mothers in the Imperial harem, who would curry favor with the Sultan, and attempt to persuade the Sultan to grant their respective sons additional power and stations close to the capital. Thus, the chief consort gained prominence, which led to a period of non-official primogeniture (as the chief consort was the wife of the firstborn son). Instead of a military struggle after the Sultan's death, the positioned favored claimant would succeed to the throne, and then generally practice fratricide against his brothers and half-brothers, usually through a ritual strangling with a silk cord. This often led to those brothers/half-brothers either rising in revolt, or fleeing the empire. Later, instead of fratricide, the brothers/half-brothers were usually imprisoned for life in the Imperial Harem, which occasionally resulted in their succession to the throne following death of Sultan.

Finally, the Ottoman Empire entered a period where the sons were kept in the Harem even prior to the death of the Sultan, and were no longer sent out for provincial governorships, in order to prevent the cultivation of rival political power bases. This had its own problems, however, as it could leave to scenarios where there were no grandsons (i.e. the succeeding Sultan did not have a son upon succession), and possibly was unable to actually have one, a condition that wouldn't be known until after succession. Therefore, the Empire shifted to a system of effective agnatic seniority, or succession of the eldest male relative, which continued to the end of the Empire.

It has been argued that the open succession led to the succession of strong, able rulers, and that the later periods of primogeniture and agnatic seniority led to weaker rules, which resulted in the decline of the Empire. I think that often lines up temporally in the sense that there were weaker rulers later in Empire's history, but correlation does not equal causation, and while it may have been a factor, there were certainly other factors at play as well.

Most of this knowledge is taken from "Osman's Dream: The History of the Ottoman Empire" by Caroline Finkel, "The Ottoman Centuries" by Lord Kinross, "Destiny Disrupted: A History of the World through Islamic Eyes" by Tamim Ansary, and "Lords of the Horizon" by Jason Goodwin. There are probably many more academic works out there that some specialists here could recommend as well.