Working with Apple Photos on both the Mac and PC tends to confuse the issue.
From an Apple only perspective, here is what I experience with Photos on iOS devices, my Macbook Air M1 running Monterey, and iCloud Photos via Safari:
First, your export to external drive query. Using Photos on the Mac, after you “Select” specific photos (as many as you want), you click Export under the File menu. It will give you some choices to export photo or original photo, Select what you want. It will then pop up a menu to select file type and name. When you click “Export” at bottom of that page, it will pop up a Finder window, where you can select where you want the files stored. As long as your Finder preferences include external drives and Hard drives in the sidebar, you can have all the selected photos exported to your preferred location directly.
Next, if you choose to use all the Apple Photos features, your iOS devices, your Mac Photos, and iCloud will sync photos, and should stay in sync with a little luck. If you take a photo with your iPhone, it should appear on your Mac a short time later. If you delete a photo on any device, you get a popup informing you it will be deleted everywhere. A couple of things to keep in mind. All devices must be logged into same iCloud Apple ID account. You need to turn on Photos in the iCloud settings on each device. If your Mac Photos is not in sync, you should run “Repair” on it. To do this, you close Photos, then you hold down the Option and Command keys at the same time, and click on the Photos icon and select Repair. After it completes repairing (supposed to be non-destructive) your Mac Photos should sync with iCloud.
Personally, I keep the management of Photos on the Apple devices, and use Google Photos on the iPhone to provide access to photos on a PC if needed. I have just proved to myself that if I delete a photo from Photos on my iPhone, the same photo is deleted from Google Photos, so be careful.