How does low cysteine conditions affect pheomelanin production?
First: Your teacher is right with the statement that a low cysteine concentration affects the synthesis of pheomelanin. To understand this, you need to take a look at the biosynthesis pathway of melanin (from here):



The first steps for both melanin forms are the oxidation of the amino acid Tyrosine to DOPA (L-3,4-Dihydroxyphenylalanine) and the to DOPA quinone by the enzyme Tyrosinase (abbreviated Tyr). In the following step the branches divert, if there is enough cysteine present in the cells, then Cysteinyl-DOPA and subsequently Pheomelanin are synthesized. 

If the cysteine concentration in the cell falls below the threshold of 0,13µM (this is taken from this paper), then the switch to eumelanin production occurs.

In the case of you second question, there is a different setting. The overall rate of the melanin production (no matter which flavor) depends on the amount of active Tyrosinase (for the production of the first steps in the synthesis cascade). The production of this enzyme is regulated by receptors in the cellular membrane (and different other factors).

In fair skinned people (classically red-haired and with freckles) these receptors are not fully functional, resulting much less Tyrosinase enzyme in the cells. This results in fewer available precursors for the melanin production and in turn in less need for cysteine. The reason is that the cysteine pool of the cell is not getting exhausted.