Why does this not work? “ls *.txt | xargs cat > all.txt” (all files into single txt document)

It's useless in the sense that using it like that doesn't accomplish anything the other, possibly more efficient options can't (i.e. producing proper results).

But cat is way more powerful than just cat somefile. Consult man cat or read what I wrote in this answer. But if you absolutely positively only need the contents of a single file, you might get some performance advantage from not using cat to get at the file contents.

Regarding readability, this depends on your personal tastes. I like cating files into other commands for the same reason, especially if the performance aspects are negligible.

It also depends on what you're scripting. If it's your own shell and convenience methods for your desktop machine, nobody except you will care. If you stumble upon a case where the next tool in the chain would be better off being able to seek, and distribute this as a frequently used piece of software on some minimal Linux system on a low-performance router or similar device with real limits on processing ability, that's different. It always depends on the context.

mkdir does not output anything, therefore xargs will not do anything useful. I don't understand exactly what you want to do, so you should explain your question better.

"would like to move a subset of files into that directory" -> do files in this subset share one or more traits? If yes, use the find command like this:

find [conditions] -exec mv "{}" dirname \;

If you want to avoid typing the name of the directory twice, or if you are doing this from a script, you can do something like

dirname=test
mkdir $dirname && mv filename $dirname