Is it possible to look up banned members?

You can look up any member who hasn't anonymised their account. Their current status will be on their info/avatar and if they are banned will be clearly shown.
I don't think there is a list of banned users.

The banlist went way before the move, as people were registering incredibly abusive usernames knowing that they'd appear in the banlist for a period of time.

I find using Google or other search engine and searching via "site:boards.ie" with username and other relevant terms delivers far better results than relying on boards search.

I can't even find my own posts on here, some of them come up but there are huge gaps.

There seem to be a few different types of closed and anonymised accounts. Some posters (e.g the poster who boasted about not changing his underpants and bed sheets) have been erased from existence. Others have "this post has been deleted", others have a scrambled username but their post count number is correct and postsremain. These can be easily searched using Google.

Certainly will not find all your posts, but if you enter the name in the 'Author' search, it will find posts and therefore your profile. Which is what the OP wanted to do.

This is what I get - 1,2k posts:

Search by author . You searched by content.

The site does have a record of them, if you start typing in the Author search box the name will come up to be clicked on.
There are no posts to view because they have anonymised their account.

Leo Fatkar's posts are still there.