Is Launchpad icon supposed to be in Launchpad?

It is not. Launchpad shows applications in the Applications folder, except for itself. There are some default categorizations, e.g., Games. "Other" contains the applications that are in the Utilities folder (within the Applications folder). If you have other folders in the Applications folder, those show up as named with (Other) appended (e.g., I use Canon cameras and I see a group in Launchpad that called "Canon Utilities (Other)" that contains the apps in that folder.

I do not see the Launchpad app when I launch Launchpad.

No. It is not normal to see the Launchpad app icon in the Launchpad space at all.

Once upon a time I had the same experience, but no longer recall what the fix for that weirdness was. Sorry.

Just as aside, I heard Launchpad has been deleted from the new Tahoe OS to be released this fall anyway.