There is a file named index.dat, and subfolders named serially from 00 to 0f, a0 to af ... f0 to ff in my `~/Applications` folder, can I delete them?
Like all things, backup your Mac and then delete the files you suspect are bad / need to know when they re-appear. Backups exist so you are free to clear things knowing you can get them back if you find out later some specific program actually needs them where they are.
Examining the files like this you have some clues that often will let you refine what caused them to be there:

What permissions / ownership is on the files?
What are the created dates, last modified dates?
Can you catch it with lsof or fs_usage (both of which are covered here on site pretty well on how to check for what process has a file open and which process reads / modifies / writes to a file)
Open terminal and use the file command to actually inspect the contents. Open them in an editor or a great free program like BBEdit to look at them safely. (Type file and a space and then drag one or two of the curious file icons into the terminal window and press return)

What’s clear, is it’s not a program that behaves at all like a mac program and puts application files where they belong in user or system libraries. I would focus on things that are cross-platform or open-source as that’s a typical pattern to shard data to directories that are named algorithmically (Apple does this in /private/var/folders) as well - but hides that away from where people expect properly curated files to exist.