I have several doubts about solving circuits.
Can any circuit be solved using Nodal Analysis?
If some circuit can be solved using Nodal Analysis, can it be solved using Mesh Analysis too?
Why do we need these techniques to solve circuits?
As long as ohms law is satisfied where V=I*R nodal and mesh can be used. They are both ways of writing a system of equations for a circuit so you know the current in each loop or the voltage at every node.
While they both work it is usually easier to solve for the unknowns with one over the other depending on the situation. For instance if you have a circuit with a current source you already know the current through that loop where as if you have a voltage source you know the voltage at that node.
These techniques give us an elegant way of finding all voltages or currents associated with the circuit which means we know if the circuit we are using in practice will be able to handle the current or voltage applied to the load.
Circuits containing voltage sources (or other elements that don't have an "admittance representation") can't be solved by the simplest form of the nodal analysis. But there is a variant called "modified nodal analysis" that resolves these issues, and allows essentially any circuit to be solved.
Just as the simple nodal analysis can't solve circuits containing voltage sources, the mesh analysis can't solve circuits containing current sources.
We need some technique to solve circuits in order to predict how a circuit will behave without physically building it. For example, actually fabricating a circuit on an IC the first time can cost millions of dollars. Before doing that, IC designers simulate their circuit to be sure they will function correctly.
Modified nodal analysis is the most common method of solving circuits because it is relatively straightforward to construct the nodal analysis equations from a netlist of the circuit, even using the computing resources available in the 1960's, when simulators like SPICE were first developed.