l will try to answer the first question in a way so that you can guess the answer of the second.
There are reasons why $\bar{M}_{g}$ is more preferred since calculations with compact spaces is easier. So l will focus on the (Deligne-Mumford) compactified moduli of curves and this involves stable curves ie curves with finite automorphism group. Finiteness condition of the automorphism group is equivalent to the condition that a curve with $n$ marked points of genus $g$ must satisfy $2-2g-n<0$ or for components $2-2g_{i}-n_{i}<0$.
Let us look at the moduli of curves with marked points at first $\bar{M}_{g,n}$. If you want to do intersection theory on this you need to find out the cycles and the characteristic classes they represents. Intersection theory involves cohomology classes(because of the connection of Poincare dual classes with the cup product).
If you consider the pairing $<,> : H^{*}(M,\mathbb{Z})\times H_{*}(M,\mathbb{Z}) \rightarrow \mathbb{Z} $ on a smooth oriented manifold $M$, it can be written as $<a,B>=\int_{B}a $ which is an integer.
We want to do something similar with the moduli "space"(actually a stack) $\bar{M}_{g,n}$ which has the dimension $dim_{\mathbb{C}}=3g-3+n$.
It has the fundamental class $[\bar{M}_{g,n}]\in H_{2(3g-3+n)}(\bar{M}_{g,n},\mathbb{Q}), $ since this is a stack or in the analytic setting a complex orbifold we should consider (co)homology coeficcients in the rational numbers.
Then the (top) intersection numbers should also be rational numbers which are
$<[\bar{M}_{g,n}],\alpha_{1} ...\alpha_{k} > = \int_{\bar{M}_{g,n}}\alpha_{1}...\alpha_{k} \in \mathbb{Q}$

Actually for moduli spaces working with the cohomology ring is quite cumbersome, instead we consider the tautological ring containing almost all the geometrically interesting classes.

For example the psi-classes $\psi_{i}= c_{1}(\mathbb{L}_{i}) \in H^{2}(\bar{M}_{g,n}, \mathbb{Q})$ where $\mathbb{L}_{i}$ is the $i^{th}$ line bundle (relative dualising sheaf actually) and there are lambda classes which are chern classes of the hodge bundle.

The point l want to emphasize here is the relative dualising sheaf. If you want to do intersection theory on the moduli $\bar{M}_{g}$ with no marked points you still need to condsider a moduli space of curves with one marked point $\bar{M}_{g,1}$ because the universal curve $\bar{C}_{g}$ in this case is $\bar{M}_{g,1}$.

You can look at Mumfords paper "Towards an enumerative geometry of the moduli space of curves" part 2, where he defined the tautological classes for $\bar{M}_{g}$ !

He considers the relative dualizing sheaf $\omega_{\bar{C}_{g} / \bar{M}_{g}}$ for the map $\bar{C}_{g} \rightarrow \bar{M}_{g}$. It is what we call the forgetful morphism of the moduli space which forgets a marked point and stabilizes the curve if needed.
In that sense studying intersection theory on $\bar{M}_{g,n}$ is a more generalized setting of which $\bar{M}_{g}$ is just a special case.