In our generation, can clothes and objects still become tamei through zera? If so how do you get rid of the tumah?
Yes, semen is still Tamei (Av HaTumah) and a lentil's worth of it would make other things Tamei (Velad HaTumah) through contact. You can remove this Tumah by dipping the clothing in the Mikvah. Why you would care to do so is not clear to me.