What happen with wavefunction immediately after a measurement?

Landau & Lifshitz use a slightly different terminology with regard to measurement than what you find in more common descriptions of the measurement postulate. In the section you mentioned, they consider the interaction between a quantum system described by a state $\Psi(q)$ and a measurement apparatus with initial state $\Phi_0(\xi)$. The interaction can be described by a time evolution operator applied to the combined system such that

$$U\Psi_n(q)\Phi_0(\xi) = \phi_n(q)\Phi_n(\xi)$$

where $\Psi_n(q)$ are eigenstates of some observable that is being measured (the state of the apparatus after the measurement $\Phi_n(\xi)$ indicates the initial state $\Psi_n(q)$ of the quantum system).

Landau & Lifshitz emphasize that in general the interaction might change the state of the quantum system from $\Psi_n(q)$ to $\phi_n(q)$ as written above. As a simple example, you might measure the momentum of particle by having it collide with a different particle, transferring part of its original momentum. Clearly the momentum state of the particle "immediately after the measurement" has changed. Note however that this does not mean that the state after the measurement "can be anything" - it means that it depends on the specific details of the measurement process.

In the more common description, a measurement refers to an idealized process where the specifics of the interaction can be ignored, essentially assuming $\Psi_n(q)=\phi_n(q)$.

In classical physics the evolution of a measurable quantity, such as the $x$ position of a particle is described by a function $x(t)$ such that if you measure $x$ at time $t$ you get the result $x(t)$.

In quantum physics the evolution of a measurable quantity is described by an observable whose value at each time $t$ is an operator. The eigenvalues of the operator are the possible measurement results and quantum theory predicts the probability of each of the possible values.

In general the probabilities depend on what happens to all of the possible values of the relevant observable because of quantum interference, see Section 2 of this paper for an example

https://arxiv.org/abs/math/9911150

This raises a problem because when I walk through a doorway it doesn't look like I have to take account of all of the possible ways I could walk through it.

Collapse was supposed to solve this problem. When you do a measurement all but one of the possible results disappears. However, collapse is not consistent with the equations of motion of quantum systems so you must either sweep this under the carpet and hope it doesn't matter or explicitly modify quantum theory. The former approach has the problem of being difficult to test. The latter approach has been followed by some physicists

https://arxiv.org/abs/2310.14969

Quantum theory without modifications predicts that when information is copied out of a quantum system, interference is suppressed: decoherence

https://arxiv.org/abs/1911.06282

Decoherence doesn't eliminate the other possible results it just prevents them from interfering. As a result, all of the possible outcomes of a measurement happen but they don't interfere:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

This is commonly called the many worlds interpretation of quantum theory but it is just an implication of quantum equations of motion applied in the same way as for any other physical theory.

A measurement in the MWI is just an interaction that produces a record and can be treated with the same equations of motion. As a result the MWI can treat repeated and unsharp measurements as well as the kind of measurements often treated with collapse:

https://arxiv.org/abs/1604.05973

Experiments to test decoherence as described in the review linked above typically involve such interactions taking place over time that could be thought of as repeated measurements. So repeated measurements without collapse have been experimentally tested. There is a large literature on such experiments such as

https://arxiv.org/abs/quant-ph/0210129

https://arxiv.org/abs/1309.1552