What is the net ionic equation?

I teach my students to do net ionic equations by the following 3 step system. It is necessary that you know or know how to work out 3 things:
How to balance a molecular equation. This includes knowing whether a reaction has actually occurred. You cannot write a net ionic equation if no reaction has occurred on mixing the two reactants.
The states of the reactants and products
Which reactants and products are insoluble in water.
Now look at what you have to do:
1) Nickel(ii) acetate with potassium bromide
Both these reactants are soluble in water.
The possible products are: Nickel(ll) bromide and potassium nitrate . Both these products are soluble in water.
Conclusion: No reaction occurs when these reactants are mixed, No net ionic equation possible.

2) iron(iii) nitrate and aqueous ammonia
Both reactants are soluble in water
Possible products are iron (lll) hydroxide and ammonium nitrate
Iron (111) hydroxide is insoluble in water . Ammonium nitrate is soluble. You can proceed with the writing of a net ionic equation:
Step1:
Write a molecular equation with states for the reaction:
Fe(NO3)3 (s) + 3NH4OH(aq) → Fe(OH)3(s) + 3NH4NO3 (aq)
Step 2:
Write a full ionic equation which shows any compound that can dissociate, as ions. Anything which is (s) , (aq) or (g) does not dissociate
Fe3+(aq) + 3NO3-(aq) + 3NH4+(aq) + 3OH-(aq) → Fe(OH)3(s) + 3NH4+(aq) + 3NO3-(aq)
Step 3:
Now write the NET IONIC EQUATION:
You do this by examining the above full ionic equation , and deleting anything which is identical on both sides of the → sign. ( in this case you delete 3NH4+(aq) + 3NO3-(aq) ). What remains is the net ionic equation:
Fe3+(aq) + 3OH-(aq) → Fe(OH)3 (s)

Now do the others in the same 3 step way:
2) barium iodide with silver nitrate
BaI2(aq) + 2AgNO3(aq) → 2AgI(s) + Ba(NO3)2 (aq)
Ba2+(aq) + 2I-(aq) + 2Ag+(aq) + 2NO3-(aq) → 2AgI(s) + Ba2+(aq) + 2NO3-(aq)
2Ag+(aq) + 2I-(aq) → 2AgI(s)
I like to simplify to:
Ag+(aq) + I-(aq) → AgI(s)

4) hydrofluoric acid with mercury (ii) nitrate
Hg(NO3)2 reacts with strong halogen acids such as HCl , HBr , HI etc – to produce an insoluble mercury salt . However , HF is a weak acid , and I do not think that it will react with Hg(NO3)2. I have done an extensive search for details of a reaction , but I cannot find any . I have only found that a possible product , HgF2 is unstable and reacts with water .
In this case I play safe and answer that no reaction occurs and no equation possible.

1)
There is no reaction, and thus there is not relevant equation.

2)
Fe³⁺(aq) + 3OH⁻(aq) → Fe(OH)₃(s)

3)
Ag⁺(aq) + I⁻(aq) → AgI(s)

4)
There is no reaction, and thus there is not relevant equation.