One gets the impression from Crisis Core and Before Crisis in particular that Turks have more clout -- whether that be actual administrative authority or just an unwritten rule -- than regular members of SOLDIER. By the same token, though, Lazard, former executive in charge of SOLDIER, had the authority to order Cissnei to spy on Zack.
So, I would say Turks "outrank" regular members of SOLDIER, but all executives outrank all Turks.
SOLDIERs volunteer for the job and Shin-Ra trains them,
Well it is mentioned that the Turks are paid better than solders, and generally speaking more money means more rank.
It's not necessarily that linear. Most equivalent professions period pay better than being a soldier, and I'm willing to be that even applies to SOLDIER. Like real world governments, higher paid positions are usually given to agents/jobs outside the military chain of command. The relatively lower (but still appreciable) pay of military personnel is offset by the numerous standard of living benefits of serving.
In a governmental aspect, more pay equals more rank within the military, yes, however more pay often just means they're not in the military, period.
Well it is mentioned that the Turks are paid better than solders, and generally speaking more money means more rank.
SOLDIER's, as powerful and elite as they are, are just that - soldiers in an army. Turks are like the secret police, special agents, 007's of Shinra. They go on top secret missions, play spy games, ferret out information, and do assassinations and other hush-hush stuff. Both are just taking orders from higher ups, but the Turks are more specialized and versatile, and more trusted with secret operations and info. Like real secret agents, the Turks don't really have an identifiable "rank", and pretty much are given carte blanche in terms of authority, as long as it falls within the scope of whatever mission they're assigned. "License to kill" and all that. SOLDIERs are just weapons, tools, muscle. Shinra points them at a target, and they nuke it.
That is true. However Shin-ra isn't a government. It is a corporation that has taken on government like qualities.
It's a corporation that became the government. There's no other way to put it.
There's what Ryvius mentioned, as well as that Shin-Ra controls the military/police. Mayor Domino is also said to have no real political power.