To what extent does the concept of "particle" in QFT correspond to the concept of "particle" in experimental physics?
Your description of what a particle looks like in quantum field theory is not accurate, but you could be forgiven for getting that impression from a first course.
To recap, quantum field theory is a quantum mechanical theory describing an arbitrary number of particles. In order to start, we need to specify what the quantum states are. This is done by constructing the Fock space, which is roughly $$H = H_0 \oplus H_1 \oplus H_1^{\otimes 2} \oplus H_1^{\otimes 3} \oplus \ldots$$ where $H_0$ contains a single vacuum state, and $H_1$ contains the possible one-particle states, and $H_1^{\otimes 2}$ contains the possible two-particle states, and so on.
For concreteness, it's useful to pick a specific basis for the one-particle states. In the case of particle physics, we're often concerned with scattering experiments where particles come in from infinity with a very well-defined momentum, so we pick the momentum basis. In this basis, every particle is infinite in spatial extent. But in condensed matter physics, you can just stick wires into whatever solid you're studying, so you can do measurements in the position basis. Accordingly condensed matter field theory sometimes introduces $H_1$ in the position basis.
There's no fundamental difference, because we can always go back and forth by linear combinations. The other crucial thing is that, starting from momentum states, we can form finite wavepackets by superposing nearby momenta. Wavepackets can move, as shown here, and they have roughly defined trajectories. This is how we model the initial states in real scattering experiments, where our collider unfortunately has a finite size due to budget constraints.
To answer your other questions:
- A particle can have plenty of other properties. For example, a particle can carry some spin or charge; this is all accounted for in $H_1$.
- Particles evolve in time, even when they're in momentum eigenstates. For example, an unstable particle will decay. Or, if you have a state with an electron and positron in momentum eigenstates, they will annihilate.
- In colloquial language the word 'particle' is reserved for states in $H_1$ that are reasonably localized in position, while 'wave-particle duality' is nothing more than the statement that some states in $H_1$ are localized and some are not. When doing quantum field theory, we just call everything in $H_1$ a particle; there is no need to split it up.
It is true that what we measure in particle detectors aren't the states you usually use to do computations in QFT. The place where your second, more intuitive notion of a localized particle pops up in the theory is when you define the S-matrix, which relates the in and out states of a scattering event. These are defined as well-separated wave-packets respectively far in the past and far in the future.
It turns out you can relate the two pictures -the theoretically convenient and the experimentally relevant- by means of the LSZ formula.