Fourier Transform Directions
INCORRECT. There is no difference between the images produced, whichever direction undergoes a Fourier transformation first.
INCORRECT. Fourier transformation along the frequency encoding direction if k-space is not faster.
You may be recalling the fact that frequency encoding takes place very quickly (for the duration of a single MRI signal measurement), whereas phase encoding is not completed until all the lines of k-space are filled. But these are statements about signal measurement, not image calculation (for which the Fourier transform is used).
A 1DFT across k-space, then a 1DFT down k-space. Note that after the first 1DFT, the column-data of the image have been correctly determined (there isn't much signal over on the left where there is no part of the knee, for example). Where the data should be within each column (the rows) is still encoded. Then the second 1DFT sorts out the row-data from within each column.
CORRECT. Once k-space has been filled, there is no difference between the data in the frequency encoding direction and the phase encoding direction. In fact, given a k-space data set, it would not be possible to tell which direction was frequency encoded and which was phase encoded (encoded by introducing different rates of change of phase over many signal measurements). The Fourier transform sees both as the same.