A state has memory. The value of a state of a causal system reflects the values of all the previous states and inputs. A linear combination of all previous states and inputs, is a linear state equation.
A measurement only reflects the value of the current state and current inputs. A measurement needs to be noisy. A Kalman Filter estimates the current state from the noisy measurement.
The presence of measurement noise is why it is called a filter. If there is no measurement noise, one uses a state observer, not a Kalman Filter. The KF equations don’t work without measurement noise.
If the measurement noise is a linear combination of the current state and input, which includes noise, the measurement is linear. When the state and measurements are linear, the KF is a linear KF.
Essentially a state has memory and a measurement only depends on the current state. We can write the state evolution as a recursive equation because it is a Markov process.
The dimensional issues are related to what older books called the controlability and observably matricies. More recent books introduce Grammians.
There are variations of the KF that include nonlinear states and nonlinear measurements. The linear KF satisfies a number of optimality criteria if the matrices and initial conditions are accurate. The other types aren’t optimal but typically good if the approximations are good.
One does not add error unless you are doing a simulation. you need to know the covariance to actually filter.
The KF can be derived a few different ways, (least squares, orthogonality, bayesian) and the solutions can be written out differently so books often don't exactly agree. I prefer the innovations form of the KF because the innovations are an online diagnostic. You can test if you have enough states or too many states.
The questions related to the actual state and measurement equations are really independent of the KF. A book on linear systems using state variables is where you learn to write those, but many KF books have a chapter or 2 on the topic. Usually, how to convert from continuous time differential equations to a set of difference equations. Writing the state equations is really a prerequisite. Most of the people who ask for help tend to fall in the category of not understanding state variables.
If you have problems understanding your state equations, you should include the specific equations. No one is watching what you see on the internet. This isn't pornhub.
Kalman Filters tend to work even if you don't know those parameters well. People tend to spend a lot of time tweaking their filters so expect to tweak a lot. The theory can be intimidating and one can often build adequate filters without a deep understanding. If it works, it works. If it doesn't work and you think it should, the theory becomes important.