The shapefile you provided seems to be [valid][1], and the projection in the prj files states (if im not mistaken) that your coordinates of each vertex are tied to WGS84 (EPSG:4326)

Using [ogr2ogr][2] [PGDump driver][3] you can translate your shapefile to a sql querry that can be used to insert data into the database: 

If you are in Windows you can open the console with `MSYS Shell`. It should be installed by default if you have used osgeo4w tool to install qgis.

Open your `msys` again and `cd` to your data directory, then for one file run: 

    ogr2ogr.exe -f PGDump -lco schema=vectors -nln name -lco geometry_name=geom -lco fid=gid -lco srid=4326 temp.sql time_1057.shp

 - the schema can be whatever you want.
 - -nln is the name postgis table that will be created. 
 - the geometry_name and fid were choosen for compability reasons.
 - the srid was extracted from your prj file.

afterwards you can use `psql` to execute the sql file in your database. eg:

`psql -h IP -U USERNAME -f temp.sql`

Note that I didn't need any accompanied database files. The created table will contain data regarding the id (field name `gid` that we choose before) and the geometry information of the shape (field name `geom`). It's up to you to create and populate more fields fields. 

Now since you mentioned that you have thousand of shapefiles, its a good idea to batch import them. Use again your Msys and `cd` to your data folder and use something like the following:

    for f in $(ls *.shp); do 
          ogr2ogr.exe -f PGDump -lco schema=vectors -nln $(basename $f .shp) -lco geometry_name=geom -lco fid=gid -lco srid=4326 temp.sql $f;
          psql -h IP -U USERNAME -f temp.sql

After the operation completes, you should have a schema called vectors and within it a table for each shapefile you had

  
NB: There is a way to insert directly your data to the postgis, but i find if you use an intermediate file to do so, is easier to understand what's happening. 

  [1]: http://www.gdal.org/ogrinfo.html
  [2]: http://www.gdal.org/ogr2ogr.html 
  [3]: http://www.gdal.org/drv_pgdump.html