"There are two pillows and a blanket on the bed."
Translation:Az ágyon van két párna és egy takaró.
If you said "két párna és takaró", that would mean "two pillows and (two) blankets". So, yes, "egy" is needed here.
Both subjects are grammatically singular, so the verb stays singular as well.
if you use direct or implied indication to the amount (here "two and one", one direct and one implied), then the subject will stay in singular form: 15 chickens are there - "tizenöt csirke van ott", or some plates are on the table -"néhány tányér van az asztalon". If you mention multiple objects without any indication of their number, then the subject will be in plural form as well (The men are there - "A férfiak ott vannak")