Your filename is kept
The converted file keeps the name you gave it, with only the extension changed. Nothing is appended, no brand is inserted and no counter is added. These files get uploaded straight into forms that sometimes check the name, and an unexpected one is one more thing that can be rejected for no good reason.
There is no watermark either, at any size, on any number of files. A watermark on a free converter exists to sell you the paid version, and we would rather not have that conversation with you at all.
Converting several files at once
Drop as many files as you like. Each one is converted separately and appears in the list with its original size, its new size and the difference between them, so nothing about what happened is hidden from you.
Download them one at a time, or use Download all to take the whole set as a single zip. The archive is built in your browser too, from the files already sitting in memory, so it costs one click and no waiting. Images are already compressed, so the zip stores them rather than compressing them again, which would only make the file larger.
If a file cannot be read, that row says so and the rest carry on. One awkward file in a folder of fifty does not stop the other forty nine.
What happens to the metadata
The conversion decodes the picture and encodes it again, so the embedded metadata does not travel with it. That includes EXIF data: the camera model, the exposure settings, the date, and on a phone photograph the GPS coordinates of wherever it was taken.
For most people that is a benefit rather than a loss. A photograph taken at home and uploaded to a public listing carries the location of your home inside it unless something strips it, and very few sites do. Converting the file removes it as a side effect.
If you actually want the metadata kept, for a photography portfolio or a copyright claim, this is the wrong tool and you should keep the original alongside the converted copy. Colour profiles are handled the same way: the image is converted into the standard sRGB space that browsers and the web use, which is right for anything published online and wrong for a print workflow that depends on a wider profile.