From this description, I believe you need a database. and this will range in complexity - I think that at the very least you may need at least 4 tables. one for users (store their data and login credentials) one for the generic game board (stores commonly re-used elements) one table for customized games, which would be keyed to both the user and the generic game. A questions table for every question entered, which has a unique key for each, and depending on whether or not it matters to your Logic - the user assigned to that question (or the owner of it) - and a table where you assign the question to the custom game.
I have no experience with using databases for this purpose. Would you think this would be rather complicated for a beginner? I run a website for teachers and a lot of them have been requesting something along these lines for a while.
The unfortunate answer is generally yes - just getting to the basics would be:
having an active server, making sure it is properly maintained and secured
creating server-side software that not only properly stores and preserves the data, but deals with user authentication and sessions
Writing a client-side communication layer to talk to this server
There's a lot of services, frameworks, and companies that can help with all of these, but that also adds another layer of knowledge - knowing what to choose best! There's also new trends in "no code" tools, but even these do require some code for what you'd need (and I'm not sure how well they handle your problem domain). The range of choices is vast.
But I do not want to discourage you -- all of this is doable, and not an insurmountable problem! Also it is often okay to not choose the optimal solution first. Most of development is trying something, figuring out where it falls down, and then with new knowledge making a better v2.
Not offering any guidance on usage with this,
But I have used Appsheet to give people access to control data as a content management system
Appsheet tries to take out the coding side for you and links in with google sheets as a database.
Auth is done in my case by sharing the google doc/sheets with them and the Apps I made with Appsheet.
On an aside my Hype sites can link to the google sheet and display the data changes they make in the appsheet app.
For example the need to change the text in a scrolling ticker tape. They just bring up the App and change it there. The website ticker tape will change accordingly.
The Hype to google sheet code is the only real coding you need and a search on the forum will show you how to get started with that side. Note that the info on google may have to be public and I would piblish it as csv and use some parsing code to parse it to json (easily found on the web). Then parse the json