What is the function of cystine, cysteine, and cysteine protease?
I'm not sure as to what level of detail I should answer this question, but I will try to give you as much information as I can that I think will be relevant to.. whatever it is you are doing.

Cysteine proteases

Are a class of enzymes that degrade proteins, and utilise cysteine-residues as part of their catalytic mechanism. 

Cystine

Is, as you said, a cysteine-dimer.. Which basically means two cysteines stuck together.

These are often formed in the intermediate steps of a cysteine protease's catalytic mechanism.

But also, during protein degradation within the lysosome, cystine can be released as a free molecule. When this happens, it will be transported in to the cytoplasm, to be processed via anabolic pathways into various products, such as glutathione (an antioxidant).



So, basically, the function of your seven-transmembrane (7TM) transporter protein is:

To make accessible to the cell, the digested products of proteins (or, amino acids).



Cystine is an amino acid. Cysteine is also an amino acid. These can all be used by a cell to create new molecules, such as proteins, or antioxidants, or vitamins or cofactors, etc., so many things. And can indeed be broken down in to cysteine.

The function of the CTNS gene, is to provide the cell with the genetic information used to synthesise the 7TM-transporter protein.