The error means that the Proj library that GDAL is using for coordinate transformations does not find a file named “proj.db”. That file is a SQLite database where all the parameters needed for the transformations are stored. Environment variable “PROJ_LIB” should point to the directory where the proj.db file is. You have some issue with your GDAL or Proj installation. If you do not change the coordinate system then the error may not affect you.
Warning means that you have some field of type 64 bit integer in your shapefile but GMT format does not supports that datatype so it is changed into datatype “real”.
You can avoid the warning by defining the datatype mapping yourself as documented in https://gdal.org/programs/ogr2ogr.html
-mapFieldType Integer64 Real.