Beats the heck out of me, but now it is working! I think there must've been a problem with the "Naim Exclusive Radio Paradise" station - perhaps connection problems... but the weirdness has disappeared. FYI, I think the problem I was having with the Celtic Moon radio station was either that I saved the playlist as "Music" instead of Podcast... that or the weirdness infecting "Internet Radio" in general was also keeping it from playing after closing/opening MMW. Also, I had NO trouble with a station in the Icecast directory - it saved and was able to be played even when I was having trouble with the "Internet Radio" stations. So. Problem solved (?) or at least mysteriously went away. Thank you for your help.

The interesting data is SO far down (Sorry, I should have had MMW come up already on Playlist, rather than Music... it did "checking" on over 3,000 songs first). You might go down to where the URL for Radio Paradise appears 37.130.228.60:8014
FYI, the previous song in the now playing section was from the Music area (not playlist area) and was Bob Dylan's "New Morning." So that is OLD data that was previously on the screen. At location 00083192 74.03091431 it appears that MMW WAS reading the URL
and received data for a new song from Radio Paradise:
00083553 87.52832031 [14036] HttpStream: Icy Data: StreamTitle='Anna Ternheim - Bow Your Head';StreamUrl='http://img.radioparadise.com/covers/l/B005MPYHJ8.jpg';

- and around here an error is visible: Again, sorry for all the excessive logging info on the top 90% of the log. Thanks.

Okie-dokie. I've got a log and sent it in under ticket: #MXR-192-36593 Logfile is log20180626a.log (well, inside a .zip)
May I assume it is OK to continue to execute the Media Monkey Debug version (which overwrite the my current version, I believe) even when no debugging is necessary? (it's not going to start creating endless, huge logs, is it?) Also when a new MMW release comes out - any thing I should/should not do? Thanks.

Yes, you need the debug version of MediaMonkey. You don't need to do anything in DBGView other than saving the log after you encounter your problem.

Oh...I went straight to 4B. Maybe I need to download and execute the MMW debug program at the top of item 4 first? Thanks.

Sorry, but the debug program's Help Manual is in the .chm format which Windows doesn't support in their newer OS's, so I'm just guessing how to operate the debugger. I opened a log file, executed MMW and clicked on a few things (tried to play the radio station playlist), exited MMW. Then browsed the playlist. It only has these entries:
00000001 0.00000000 [7792] ASL checking for logging parameters in environment variable "MediaMonkey64Helper.exe.log"
00000002 0.00026819 [7792] ASL checking for logging parameters in environment variable "asl.log"

Tried it a 2nd time, and did an additional "save as" before closing down the debug program, and still only got the same thing.

Never mind, I found it in the "How to report a bug" Forum.

When I click on the link (or copy & paste it) I get "The requested topic does not exist."

Capture this in a debug log (step 4b) and attach the log to a Support Ticket: http://www.mediamonkey.com/forum/viewtopic.php?f=1&t=69

I've waited as long as 17 minutes. I've noticed something else. When I initially double-click the "track" or alternatively hit Play, the play button immediately changes to the Pause key for 3 seconds, then changes back to the Play key. Any secondary attempt clicking on the Play button changes to Pause and then back to Play in less than a second. (In neither case is any sound produced). I've also noticed that during the initial 3 second "playing," at the bottom line on the screen, the full "song title" of "Radio Paradise - Naim exclusive" appears, after which it changes to solely "Naim exclusive." (Not sure if that has any bearing on anything.)
Moreover, on the 2nd radio station I put into another playlist (Celtic Moon), the headers & columns are different. It appears that Celtic Moon is treated as an Album/Song (ie: there are columns for Artwork and Album) whereas these are lacking in the Radio Paradise radio station. However, when attempting to play the Celtic Moon radio station (and each time, this is after a closing of MMW and restarting of the app) there is no 3 second delay, but an immediate change back to the Play button. Other music and playlists work ok. Now, SOMETIMES after "playing" the Radio Paradise station, nothing else (neither regular music tracks or other regular playlists) will play - nothing will play and I have to close MMW down and start it up again. Incidentally, when I close MMW, after the screen goes away, after about another 5 seconds, I get a sort of "notification" rectangle in my lower right corner of my monitor with a browninsh-gold "disk" image (like blank artwork), plus the Radio Station name and a song name that I suppose was "playing" although I could not hear it at any point). Also, sometimes, after changing/executing too many things, the app just locks up and I have to kill it using Task Manager, but I can't seem to recreate that on purpose. There's so many weird things with this process that I may just have to give up on trying to save favorite radio stations. I DO appreciate your help. Let me know if you want me to try something else. Thank you!

Multiple Radio Stations won't work as it is a continuous stream. So you can store them in the same Playlist, but once one is playing it should play forever.

How long did you wait after starting playback and have you tried using Play again.

Not sure what I was doing wrong... Media Monkey completely locked up 3 times (had to use Task Manager to kill the app), but I finally got it to work and have saved the station as a "song" in a playlist. I then renamed the song Title that was playing to "Radio Paradise." I THINK there are a couple of alternatives here, and please correct me if I am wrong. I am guessing that I can do the same thing with the next radio station I find (turning off Random play), OR I could just rename the entire playlist to "Radio Paradise" and just keep 1 "song" (Radio Station) in it... and add a new radio station into a new playlist every time. Well, I was going to say the problem was solved, but upon closing MediaMonkey and re-opening it... my station is still there in the playlist, but when I click on Play, the Play button changes to the Pause button for a second (as if it is playing), but then changes back immediately to the play button. IE: nothing plays. I've started all over and even re-captured another station in a new playlist - which plays, but when I close MM and go back, I have the same non-playing problem. I've tried changing the options from random or continuous play, etc, but nothing seems to help. Any ideas? Thanks!

1) File > Open URL enter http://37.130.228.60:8014/
2) Right click on the entry in Now Playing and use Send To > Playlist to add it to a Playlist

An addendum to the above reply. Closing and restarting MM fixed the "not able to play anything" glitch.
FYI, I looked up my default .Mxx applications using the Win 10 Settings screen. Here's the default media types I found by application:

Windows Media Player: .m1v (Movie Clip), .m2v (Movie Clip), .m3u (M3U file).
DBDFab Media Player: .m2p (M2P File), .m4b (M4B File), .m4p (M4P File).
Movie & TV: .m2t (M2T File), .m2ts (M2TS File), .m4v (M4V File).
iTunes: .m3u8 (M3U Audio Playlist [UTF-8), .m4r (Ringtone).
Groove Music: .m4a (M4A File).
Media Monkey: .mac (MAC Audio File).
- I think I should change .m3u8 to Win Media Player. Suggestions? Thanks!

Ah, I needed to read your suggestions more closely. Yes, when I go to the top MM File/Open menu, there is an "Open file or URL" option. When I select that and paste in your URL (http://37.130.228.60:8014/) it does play in MM, and in fact, the current song playing even appears in the "Now Playing" column on the far right. However, I cannot click and drag that "now playing" song to the playlist (if that was your intention). Ok... so, now to "reset" things, I went to my music option and played a song from my library, thus ending the radio session. Then I went back to Internet Radio / searched radio paradise / and clicked on M3U (which I had never done before, having previously always clicked on the underscored radio station name). Interestingly, it immediately executed Windows Media Player (not giving me the option to select a different player) and started playing the Internet radio station (But in the "Now PLaying" far right column of MM, it still showed the last song that I had played from my library). Then I right-clicked and went "back" to the previous screen and instead of clicking on M3U, clicked as I had previously done on the underscored Radio Station Name itself, and as before it came up with the various icons of media players to select for execution. Ultimately, though, I am not seeing anything that can be dragged/dropped from the "Now Playing," far-right column in MM to the playlist section on the far left column. Perhaps you're dragging something from somewhere else? Oh, and how did you obtain the IP address (37.130.228.60:8014) for Radio Paradise? (And can I assume I could have just entered their non-IP address of http://www.radioparadise.com/ in the File/Open URL option?) Uhoh... I went back and forth between my music library and internet Radio too many times, I guess. Nothing will play now, neither songs nor radio. I'll close down MM and restart it and then I guess I will also check out http://defaultprogramseditor.com/ and see what it's about. Thanks - I'm learning a lot here!