Reversibility of a reaction
Strictly speaking, the answer depends on conditions and, as a consequence, states of aggregation.

Reaction (1) can be reversible if iron(II) sulfide is subsequently heated above $\pu{700 °C}$ in vacuum to prevent oxidation of the elements [1, p. 422]:

$$\ce{FeS(s) -&gt;[\pu{700 °C}][vac] Fe(s) + S(g)}\tag{1a}$$

Reactions \eqref{rxn:2a} and \eqref{rxn:3a} are indeed irreversible if carried in aqueous solution — as you already mentioned, a poorly soluble salt is formed in both cases:

$$
\begin{align}
\ce{NaCl(aq) + AgNO3(aq) &amp;-&gt; AgCl(s) + NaNO3(aq)}\label{rxn:2a}\tag{2a}\\
\ce{H2SO4(aq) + Ba(OH)2(aq) &amp;-&gt; BaSO4(s) + 2H2O(l)}\label{rxn:3a}\tag{3a}
\end{align}
$$

Reaction \eqref{rxn:4a} is a canonical example of a reversible reaction used to study equilibrium in a gas phase:

$$\ce{H2(g) + I2(g) &lt;=&gt; 2 HI(g)}\label{rxn:4a}\tag{4a}$$

I'd say it's impossible to tell in advance what reaction is reversible or not without conducting an actual experiment or at least estimating free Gibbs energy.

References


R. A. Lidin, V. A. Molochko, and L. L. Andreeva, Reactivity of Inorganic Substances, 3rd ed.; Khimia: Moscow, 2000. (in Russian)