There are no tariffs between member nations.
True!
The EU has twenty members.
False, it's 27.
The dollar is the common currency of the EU.
False, it's EURO
Member nations give up some sovereignty to belong to the EU.
Yes, this is true. They must obey also the European law and the respective governments can be "reprimended" by the EU Parliament
EU members speak the same language.
False, each country has their own language
Citizens traveling between EU nations do not need to present their passports at the border.
False, but partially true: only between the Schengen-States. Romania, Bulgaria, UK and others still have borders with passport controll
Citizens of EU nations can live, work, or study in any member nation.
True, but there are some restrictions for new member states.