Linear dependence on photoelectric current

I have done a lab experiment in which we had to determine the Planck's constant through photoelectric effect. The equation reigning this experiment is:

$$eV_c=h\nu-W$$

Where $$\nu$$ is the radiation frequency, $$W$$ is the work function of the material and $$V_c$$ is the cut off voltage, i.e. the voltage to which we get no current. The experimental results I get are somewhat expected, that is, the current decreases as the stopping voltage increases.

I was told that, theoretically, current should vary linearly with the stopping voltage, and so I made a linear fit in the region that best resembles a straight line in order to find $$V_c$$. And repeated this process for several other light sources so as to mage use of the equation stated above to determine $$h$$. And this method is also used by others. Please see this and this.

But I also noted that, both the results from the methods mentioned in the above links and mine show an error of about $$50\%$$, that is, the value of $$\text h$$ determined is about half. There are several things I don't understand:

i) Where is it stated that this linear dependence should be verified? As far as I know from a bit of research predicting photoelectric current is extremely difficult and "not worth the effort".Please see this.

ii) The equation stated above is based on a monochromatic light source, but here I am dealing with a continuous spectrum, although it has a narrow peak at the most intense frequency, it still emits photons of about every frequency, so how should this data be interpreted? If I find $$V_c$$ trying to lower the current until $$0$$, then isn't this cut off voltage correspondent to the more energetic photons and not just those that are predominant in the light source, near its peak?

iii) How can one explain getting almost half the Planck's constant, and the similar results obtained by more than two people? The method clearly gives consistent but wrong results, but why? Is it too much of a coincidence?