The songs I downloaded online do not show up with the same name in the player
Google Play Music is most likely reading the MP3 tag instead of the actual filename. There are two ways to edit the MP3 tag. 

To do it on your computer, I have used in the past mp3tag. To do this from your phone, you can use the application iTag - Music Tag Editor. If you prefer neither of these methods, you would have to find a music player that does not read the MP3 tag but instead reads the filename. I use Poweramp and in the settings of that application, under Look &amp; Feel &gt; Player UI, you can set the player to read the titles from the file names.