Photo organization on iPhones:
Older iOS iPhones use the 100 Apple and 101 Apple.... folders. Usually the newer pictures are added to the highest numbered folder, but when older photos are deleted, sometimes the new ones are added to the lower numbered photos.
The reasoning for doing the multiple folder organization is storage rules. The iPhone needs to provide access to those folders over USB, and a specific file system is expected by PCs and Macs. There is a limit to the number of files that can be stored in a folder in that file system. So when the number of photos exceeds that number, a new folder is created.
Newer iOS changes the organization a little bit. The folders are named by year and month making it easier for humans to find the files they are looking for (shocking to see Apple change something like this to help the user, they seem to like to be as backwards compatible as possible).
Backups and seeing pictures on the PC:
Backups with iTunes does have all the photos in them. They are NOT easily accessible to you. They CANNOT be easily viewed. If the backup is not encrypted with a password, third party software is needed to read the backup file, and extract the photos. I discussed some software for this in another thread in the last few years.
I use other software to copy (or copy and delete) pictures from my phone to my PC for organization and storage. I use Adobe Photoshop Elements Organizer (part of a package) that usually works, but seems to struggle when there are 5000+ photos on the device, this software used to be reliable, but Adobe is either getting sloppy, or Apple / Microsoft is making it difficult to interact with other devices just so you will use ONLY their proprietary hardware, operating system and software.
SyncIOS is a 3rd party software that can get photos off of the iPhone into a folder and it seems to be reliable with large numbers of photos and automatically converts HEIC (the new high efficiency format that iOS is using when the Settings->Camera->Formats is using the default of High Efficiency) to JPG on the PC since Windows does not support this format by default (Microsoft charges $1 for the privilege of seeing this format) and most software (like Adobe) does not support it. All that can be done for free. Though there are some downsides. Pictures are not automatically rotated (Adobe does this when getting it from the iPhone directly, this does not seem to be a paid feature of SyncIOS) and all the pictures are dumped into a single folder unorganized. I make SyncIOS put the pictures on an SD card in my PC and use Adobe to move the pictures from the card reader (Adobe thinks it is a camera card) into an organized library where I tag photos (getting around the Apple/Microsoft/Adobe incompatibility issues).
iPhone album use:
First off when you create an album (Tap the + above Albums, tap New Album, give it a name and tap Save) it asks which existing photos you want to add. Tap any (or no) photo and tap Done to finish adding.
Now if you take a new photos and go into your Camera Roll to see them, you can tap Select (top right) to choose 1 or more photos. Tap the Share button (box with the up arrow) then tap Add to Album, you will see a couple options. First is to create a new Album, which if tapped will go through the same process as I described above. Under that will be My Albums, which will show a list of ALL albums that exist on this iPhone. Tap the one created to organize the photos and they get linked there.
Your process should work too where a photo is viewed, then the Share button is tapped, then tap Add to Album, find the album under My Albums, tap the one created to organize the photos and they get linked there. The photos are NOT duplicated, they only exist once on the phone but can appear in multiple albums and shared albums as you need.
What I think you are missing is seeing your albums below the new album button. This is usually caused by increased font sizes on the phone and needing to scroll down to see it.