Wavepackets 

Think of it as saying we measure the packet at x =0 (the barrier) happening at t = 0. So negative time is just what the packet was dooing before reaching the barier and positive is after.

It means that the first derivative is zero

it means that the observables are time-independent

"Stationary states" are eigenstates of the Hamiltonian operator. These are states in which a measurement of energy returns a well defined value of energy. They have a funny behavior that the observables of such states have constant expectation values. They are fictitious idealizations that are really non realizable. Any system in a stationary state will remain in that state indefinitely. However important dynamical states can be created by a superposition of such states. Calculation of these expectation values will result in interesting time behavior. Stationary states though are useful basis states for the construction of wave functions. There are numerous problems in all QM textbooks demonstrating their importance. I would have been happy to explain it but its a little hard without using a blackboard. However im creating an online course 1000 solved problems in quantum mechanics . Still working on it and I explore the notion of stationary states from different perspectives

It is the f(K) from the Fourier transform

You can think of it as a weight for the solutions , for every k

f(k) is just an amplitude function that weighs the different plane waves or stationary states in creating a superposition .