I`m living with single mom and her 8 years old son, If we separate, Can the mother sue me for child support for her child, despite that I`m not his biological father ?

She might be able to.

It will depend on how long you 2 were living together, plus where you live.

Other factors are have you been supporting her and her son, has she been working, etc?

I know in Ontario, it is possible, if certain other circumstances are met as well.

Talk with a lawyer for an accurate answer.