Doubt 1: If two particles are identical, you can not distinguish between them.
Identical does not mean "we cannot distinguish them". There is a reason why identical particles are sometimes mentioned as distinguishable and sometimes as indistinguishable.
In order to explain the difference between identical and distinguishable, let us consider simple example. Two coins from the same batch can be called identical; it means that all their intrinsic properties (weight, shape, material, minted pattern) are the same, so we cannot distinguish them based on these properties alone. However, we can still distinguish them based on their different position and orientation with respect to other bodies in the observed space.
The same goes for electrons; because all electrons have the same mass, charge and other intrinsic characteristics, they are all identical. The electron that oscillates in cellphone network tower is identical to the electron that oscillates in a mobile phone. But these two electrons are distinguishable, because they have different location.
If we discuss two electrons in an atom, we can no longer track and distinguish two electrons in such a way. That's why they are treated as indistinguishable. Even if we knew density function for presence of an electron in 3D space around the atom $\rho(\mathbf x)$ (which is a very detailed information that we do not have), we could not use it to distinguish between the two electrons.
Then, I think, permutation operation is meaningless.
Because you can not distinguish them, how can you tell if they are permuted?
In an atom the electrons are indistinguishable, so indeed it is not possible to "take them and interchange them" in the usual sense (for example, interchange their state of position and momentum). However, in the expositions of the theory of atom and molecules, the permutation operation discussed is not done to the electrons, but to arguments of the $\psi$ function.
Let the two arguments be $\mathbf x$, $\mathbf y$. These are distinguishable because they have different name, so we can do the permutation and introduce two different functions:
f(\mathbf x,\mathbf y) = \psi(\mathbf x,\mathbf y),
f'(\mathbf x,\mathbf y) = \psi(\mathbf y,\mathbf x).
Sometimes it seems not necessary to (anti)symmetrize identical particles, and people just use the half-cooked picture for granted. why?
The requirement of antisymmetrization and symmetrization is useful in determining the solutions of the Schroedinger equations for small systems such as atoms and molecules. It certainly is not useful for description of large macroscopic systems such as electrons in cyclotron. This shows that applicability of the Schroedinger equation and the usual ideas around it is limited. This is probably related to strength of interaction; interaction of two electrons separated by large distance is much weaker than electron-electron interaction in a single atom.