There are two distinct things to address:
- the album artist in your first screenshot
- the two Sort As fields in your second screenshot
The first one: the Album Artist in your first screenshot.
That's the field that's causing the artist name to the right of the album artwork to say Beach Boys, The. If you have tried changing the album artist field but it hasn't stuck, that suggests that the song, or one of the folders in the song-file path, is Read-only. To test this out, check any one song that is causing the issue - don't change anything for now:
- in iTunes, right-click a song (that's causing issues) and click on Show in Windows Explorer
- in the pop-up that appears, look on the General tab at the Read-only attribute

- if it has a tick in the box, then it's set as Read-only, which is why you cannot change it in iTunes
- if the song is not set to Read-only, then you now need to check each of the folders in the path of that file to see whether any of them are set to Read-only
- you may need to confirm that all songs on an album are set the same way - i.e. all Read-only, or all not Read-only. If an album has a mixture, iTunes will almost certainly treat the complete album as Read-only
- I am not sure what it means if there is a black box within the Read-only box. In my folders, I have several folders with a black box, but it doesn't appear to have stopped me altering the song's tags (fields)
- once you identify where the issue is, then you can change it. From what you've told us so far, I gather this is affecting several songs/albums. In that case, the issue might be:
- one or more folders in the path set to Read-only
- both a folder and all the files set to Read-only
- all files on an album set to Read-only
You will have to work that out for yourself.
Once you identify the exact folder or files causing the issue, tick on the relevant box to get rid of the tick.
The second one: the Sort As fields in your second screenshot.
By default, iTunes does not sort by the word the, or a leading A, and iTunes will automatically drop either term from the Sort As box, even though it remains in the main box. As you will have noticed in your second screenshot, the Album Artist/Sort As moves "the" to the end of the text, but in the Artist/Sort As, it removes the word "the". That's how iTunes works, you cannot change its behaviour.