Why do these reactions not occur?
When put in water a mixture such as $\ce{HCl}$ and $\ce{Pb(NO3)4}$ goes to complete ionic dissociation. Lead nitrate is soluble in water and hydrochloric acid is a very strong acid. Nitric acid is also strong, which means those dissolved nitrate ions are not reversing back to the molecular form.

Their ions will be fairly stable in solution and will only combine in case a more stable compound is formed. You could come up with reactions in which some compounds will go out of solution (like $\ce{Cl2}$), such as $\ce{ 2HCl + HNO3 \to HNO2 + Cl2 + H2O} $. Lead could also act as an oxidizing agent. But I would argue that if you consider the change in free energy or the net redox potential of those reactions, you'd find that they are quite unfavourable. If you want to calculate it itself, you could start from a reduction potential table like this one.