Q: When deleting duplicates, I usually have to delete the duplicate song twice to make it go away. Why?

It sounds like you have an iCloud Music Library, from subscribing to iTunes Match or Apple Music (or both).  The first time, you are probably removing the local copy of the song file (stored on your computer's storage).  But the song remains in your iCloud Music Library, so it appears in your music library list with the cloud with down-arrow symbol (which you can click to download it again).  You can still play the song by streaming from iCloud.  The second time, you are deleting the song completely, from your iCloud Music Library.

You can see the distinction if you right-click a song (that is stored locally) in iTunes.  On the contextual menu that comes up, there are two separate commands, Remove Download and Delete.  Remove Download removes the local copy of the song file, but keeps the song in your iCloud Music Library.  Delete deletes the song completely, with one action.

Actually, if the song is a purchased item from the iTunes Store, it may appear in your iTunes music library even if it's not stored locally.  It was added to your iTunes library at the time of purchase, and you have the choice of storing it locally, or streaming it from iCloud.  So, you're NOT subscribed to iTunes Match or Apple Music, IF the song is a purchased item.  Sorry for any confusion...

I think you'll still see the separate right-click commands for Remove Download and Delete, for a purchased song.  For a song that was added from another source, like a music CD or Amazon, you should only have a Delete command.  In EITHER case, if you Delete the song, it goes should no longer appear in your iTunes music library.  But you can later download it again (for a purchased song) from the iTunes Store Purchased screen.

There may be a setting in iTunes preferences General pane, for showing purchased songs that are only stored in iCloud.  My settings are different, because I subscribe to iTunes Match.  I have a setting there for enabling iCloud Music Library.  If you have an equivalent setting (that applies to showing purchased items in iCloud), it may worded differently.

How would I NOT store it locally?  When I click on purchase, it says, "Downloading"

For example, if you buy a song on your iPhone, it may appear in your iTunes music library on your Mac (with the cloud symbol next to it).  You can also Remove Download for purchased songs stored locally, to free up storage space, while still keeping the song in your iTunes music library, where you can stream it from Apple's servers (iCloud) to play it.  That's what happens when you try to "delete" a song, and it remain on your song list with a cloud symbol next to it, meaning it is now only stored on iCloud.

You can check for any subscription by going to your Account Information screen.  Click on your name at the top of iTunes window, and select Account Info.  On the Account Information screen, look under Settings to see if you have any Subscriptions.

Sometimes, but not always, when I do the second delete, it will say, "Hide from iTunes purchase" I don't understand this either.

You can hide past purchases, if you don't want them to appear in your Purchased screen.  First, here's how to download past purchases from the Purchased screen

Download your past purchases - Apple Support

If you don't want specific purchased items to appear on that screen, you can hide those purchased items

Hide and unhide purchases in iTunes or iBooks on your Mac or PC - Apple Support

That option (that comes up when you delete purchased songs) is probably asking, since you no longer want this purchased item to appear in your iTunes music library, do you also want to hide it from your Purchased screen?

The cloud symbol can show if you delete the local copy of a purchased track. In some cases iTunes can corrupt some or all of the database after a crash or during an update. If this happens you may see cloud links for content that is already present on your hard drive. If you intentionally delete a purchased track from your library you should first be asked if you also want to remove the attached file. You should also be asked if you want to hide the purchase from your history. If you don't do this it will show with the cloud link. Occasionally iTunes can get confused when matching up your purchase history and your local media, particularly if you make changes to metadata. In such cases signing out of the store and back in again usually resolves things. Check some of the folders where the 200 recently downloaded cloud tracks would be stored. Perhaps you have multiple copies in the folders but not iTunes.

From the sound of things you have automatic downloads enabled. This means that purchases you make in iTunes will automatically download to your device. Without this feature enabled new purchases can be made to show on a device with similar cloud links that allow you to either download to the device or stream. Streaming plays the content directly from Apple's servers to your computer or device without downloading a copy that you can play again later. Perhaps useful for those who have lots of bandwidth but not enough storage.

iTunes match extends some of the cloudy aspects of purchased content to media ripped from your own CDs or purchased from other sources. It can also be used to release DRM locked iTunes music purchases that might have been made prior to September 2009 when Apple switched from 128k DRM AAC to 256k non-DRM AAC.

tt2

I never do this, I ONLY purchase my music on my Mac Pro Tower.

But useful for people who DO buy songs on iPhone, iPad, iPod touch, secondary computers, Apple TV...

I don't need or want any streaming at all.  

But useful for people who DO stream songs.  I have a second (older) Mac connected to my HDTV and sound system in the living room.  I subscribe to iTunes Match, so it accesses my music library from iCloud and streams songs.  I also access and stream my iCloud Music Library from my iOS device, which only has 16GB of total storage; it does not have any songs stored locally.  Just because you don't use a feature, there's still a reason why it exists... 

when I buy a song on my Mac it magically shows up on my iPhone. But I don't think that is "streaming", correct ?

Depends on your settings.  If the song on the iPhone has a cloud with down-arrow next to it, it is streaming when you play it. 

I'm not really sure why that function even exists. When I purchase a song on my Mac, it immediately downloads and is stored locally.

It's equivalent to buying a music CD, and then throwing it away.  For example, I download a free song on the iTunes Store, mostly because it's free, and decide I hate it.  It's on my list of purchased songs (for my Apple ID) forever, but I can "hide" it so that I don't have to see it anymore.

Song files can become corrupted, or you might accidentally delete them, or your computer's hard drive can fail.  You should have a backup, but the features you find not useful allow you to get your purchased songs back from Apple's servers (iCloud) in a convenient way.  That's what you did with those "about 200" songs.