Is there a minimum start altitude for autorotation landings?
What you're looking for is the height/velocity curve, some times called the 'dead man's curve'. This shows the altitude and speed requirements for a safe autorotation, plus the height and altitude conditions under which an autorotation is probably not possible. 

A lot of variables go into this, and the h/v curve is specific to the particular helicopter, and in particular, whether the helicopter has a high inertia rotor system or low inertia rotor system. 

For example, the Robinson R22 has a low inertia rotor system, meaning that the rotor blades are fairly light, and thus don't have a lot of inertia to remain spinning in the case of engine failure. A similar size helicopter, the Enstrom F28, has a high inertia rotor system, so the h/v curve for the Enstrom looks quite different from the Robinson, but is somewhat similar to another popular high inertia helicopter: the Bell 206 JetRanger. 

As the article notes, Robinson doesn't recommend a climb out under 45 knots speed, while Enstrom allows it at 17 knots. This limitation is governed by the h/v curve, not performance. While the Robinson could climb out at lower airspeeds, it wouldn't be safe in the event of engine failure. 

Disadvantages of a high inertia system are that it uses more fuel, and is slower to accelerate under power in low rotor speed conditions (which are also potentially deadly).