Are Fast Wavelet Transform coefficients constant if the input signal frequency coefficients are constant?
[Specific answer] If you have a real sine or cosine, the discretization and finite support may entail issues in the possibility of having a constant: edge samples or periods that cannot be divided by powers-of-two can affect the  wavelet coefficients.
But more important: let us suppose the discrete sine period is well chosen. Then a low-pass or high pass filter on a sine will yield something close to a sine (because complex exponentials are invariant under linear systems). So even after downsamling, the coefficients in a wavelet subband are likely to behave as sinusoidal values.
If you expect &quot;constant values&quot;, you are likely looking at a complexified and absolute-valued representation of the coefficients.
With a real signal and discrete wavelet, one will likely see ripples on several subbbands, instead of a constant.
[Generic answer] The purpose of most transforms (whether fast or not) is to help in handling or interpreting the content of some data, or classes of data. Invertible transformations keep all information, sometimes with redundancy.
Discrete wavelet transforms (DWT) are meant to capture information in a non-redundant way from not-so-stationary signals. They often  are really bad as capturing information from a purely periodic sine signal.
As most wavelet filter bank filters are not perfect, and the signal is of finite length, filtering and representing coefficients will suffer artifacts.  So in a DWT, a sine would spread on many subbands, and suffer from aliasing, which is well known. However, despite the above caveat, the magic of wavelets can however recover the original sine, from the spread and aliased coefficients.
So :

don't expect DWT coefficients of a sine to be constant
don't expect them correspond to the true frequency content of a signal

They are not meant for that. However, if a signal has constant frequency content, but phase shifts (thus non-stationary), then  wavelets (maybe continuous once) can be great at detecting their location or onsets.