All resampling methods have advantages and inconvenients.
Nearest neighbours preserves the pixel values but might duplicate or remove some of them. Therefore you will probably end up with some discontinuities on your profiles (slope, etc).
Cubic convolution, on the opposite will yield smooth slopes that are nice for hydrological model, but will not preserve the pixel values. For DEM this is what I use most of the time.
Bilinear resampling is some kind of compromise between pixel value preservation and smoothing (if you can't choose).
BUT, what I recommend in your case is to keep the original CRS of your raster and reproject your vector layers. Contrary the raster projection, you do not lose information when you project your vector data, so you can return to their original CRS after running your analysis. (The only thing to care about is to densify your lines if you have very large extents, but this is usually not a problem for local analysis.)