As a cook, am I allowed to cut and cook pork?

As a cook, yes, it is permissible. You could do it. But you ask if as a Muslim is it good and should you be working in that type of circumstance? I would say 'No'. Better that you work in a Halal restaurant etc. But cutting or cooking a pig, technically speaking, what is haram is only the eating of the pig. That is what haram is clearly.