"Ik heb het varken niet."
Translation:I do not have the pig.
Niet means not as in: i do not have the pig and geen means none the sentence would then mean that you do not have any pig.
Think of 'geen' as meaning 'not a'. Using this logic, 'geen' must be used when you want to say 'I don't have a pig' - 'ik heb geen varken' - thinking of 'geen' as meaning 'not a', then this can be translated as 'I have not a pig'. But when you want to say 'I don't have the pig', you can't use 'geen' because saying 'I have not a the pig' is incorrect. So you have to use 'niet' instead, which negates the verb, the having, not the noun, the pig. So 'I don't have the pig' will be 'ik heb het varken niet' - 'I have the pig not', if you will.
Varken = pig
Big = young pig
Zeug = Female pig
Beer = Male pig
Varkensvlees = pork
Sure, though it can also be used to refer to a male pig (boar) (https://www.vandale.nl/gratis-woordenboek/nederlands-engels/vertaling/beer). Though without any further context it's same to assume it means bear.