Hello gregger44,
We understand you have questions about managing your Apple Music playlists online and on your iPhone. We're happy to give you some information on this.
Some of the options you would see in iTunes on your Windows PC, or in Apple Music on your iPhone, are not going to be available on music.apple.com. Here is some steps and information on each one of the items you listed.
Merge playlists: There isn't going to be an option to merge two playlists together into one playlist in Apple Music, however, you can take all of the songs from one playlists and add them to another. When doing so both playlists will still be active. You can then delete the playlist you no longer want. To do this, open a playlist in Apple Music on your iPhone and select the more button (three dots). After that, select "Add to a Playlist" and select the playlist you want to add those songs to. These options aren't available in the online version of Apple Music.
Move songs between playlists: Moving songs between playlists works similarly to the steps above for merging playlists. You can add a song from one playlist to another by selecting the more button, and then selecting "Add to a Playlist". This will not remove the song from the first playlist, but you can delete the song from that playlist afterwards if you choose to. This is another option you will not find in the online version of Apple Music.
Delete playlists: To delete playlists, open the playlist, select the more button, and then select "Delete from Library". You can delete individual songs from playlists you've created by taping on the more button the same way. Here's more on deleting music from Apple Music: Delete music, movies, and TV shows from your device. You will see the same options for playlists and songs within playlists when viewing them on the online version of Apple Music as well.
View additional suport for Apple Music here: Music Support.
Thanks for using Apple Support Communities.
Take care.