Warp a shapefile from transverse Mercator to Lambert projection

Question

I'm quite a newbie in QGIS. I imported in QGIS a shapefile of a European country (as well as a TIFF representing the contour). These files are referenced in EPGS 3035.

I would like to

1. transform a copy to a transverse Mercator projection (e.g. EPGS 3043)
2. transform another copy into a Lambert projection (e.g. EPGS 3034) then overlay the 2 results and see how much they differ.

I tried to use (with the raster layer) Raster > projections > reprojection but I couldn't obtain any result: the error message says that

ERROR 1: The transformation is already "north up" or a transformation between pixel/line and georeferenced coordinates cannot be computed

My final target is to obtain a graphical confrontation between Mercator and Lambert projections.

Answer 1

About the error, it is not possible to determine what is due if you do not attach a link to the file and copies the gdalwarp code used, or a screenshot of the GUI.
Likewise, you can directly export your layers, either vector or raster, to a new file with a new coordinate system, and QGIS is responsible for reproject it.
I started with a vector layer in WGS84 coordinates, I took Germany as an example and selected some grids and some countries (it would be better if you provided this information and data in the question).

Export your vector (and raster data if needed) to new files in EPSG:3043 coordinates. In addition, assign the project the same coordinate system to represent on the screen.

Then, export the EPSG:3043 polygon to a new file in EPSG:3034 coordinates. Load it on the map. Will see the polygons overlapped because QGIS is reprojecting the new layer EPSG:3034 to EPSG:3043.
But here's the trick: Change the new Layer Properties in the Source - Coordinate Reference System to EPSG: 3043. The result is that QGIS now understands that the shape has its coordinates in the EPSG:3043 system, and reprojects it as such, showing in the canvas the coordinates that the polygon actually has.

Finally, Translate (from the Vector geometry menu in the Processing Toolbox) the new polygon to some point of coincidence to be able to compare them (I used the coordinates of a vertex located on the Insel Fehmarn).