Nitrication

Yes, both ammonium (NH+) and nitrate (NO3-) ions are available to plants.

We have discussed three types of fixation processes during the discussion session #3. In this course we mentioned N2 biological fixation (that converts N2 from atmospheric air to plant available forms through activity of N-fixing organisms), NH4+ fixation (which represents trapping of NH4+ ions in the inter layer space of certain phylosillicate minerals, reducing availability of this form of N to plants) and phosphate fixation (a chemical reaction with either Fe/Al or Ca ions which leads to formation of insoluble phosphate compounds that are not available to plants). Hence, the outcomes of these 3 types of "fixation" are different

Do we need to know the equations for ammonification and nitrification?

You need to know to describe those reactions (or any other chemical reaction mentioned in our lectures). Just knowing to write chemical reactions without knowing to describe what is going on is not what we are after in this course.