What are conditions for the existence of a critical value (for a phase transition)? Can there only be a critical temperature if there is some natural unit for an observable in the model, i.e. if there is a natural scale for something? Otherwise I don't see how for a system there could be a rule how the value of $T_c$ gets actually terminated. And do this transitions vanish from the model, if one does a limit where these units get irrelevant?
Is it a priori arbitrary which quantity (lenght, energy, charge,...) has to have the specified unit? And/Or have there to be more such units to make up a critical temperature? Lastly, can there only be a phase transition in a system if there is an associated critical temperature $T_c$? 
 A: Phase transitions are abundant and occur also in athermal systems. For example, a collection of hard spheres will undergo a phase transition from a liquid-like state to a solid-like state when the volume ratio of the spheres is larger than some value. A graph with random links will undergo a phase transition if the average degree of a node is more than 1. Another good example is percolation.
So clearly, phase transitions can occur also in systems where energy plays no role. However, when a phase transition occurs as function of temperature, it is clear, on physical grounds, that the critical temperature will scale as some basic energy scale of the system. The problem is that usually you have quite a few energy scales.
As to your last question - phase transitions might occur when you change any of the control parameters. Take the liquid-vapor-solid phase transition for example. The phase space is plotted in this image, courtesy of wikipedia:

According to the current parameters $(T,P)$, your system is at a given point in this phase-space. When you change either of these parameters, you might cross a boundary between two phases - this is a phase transition.
