Why does the gzip version of files produce a different md5 checksum

You can just do:

gzip *

gzip will tell you it skips the files that already have a .gz ending.
If that message gets in the way you can use:

gzip -q *

What you tried did not work, because gzip doesn't read the filenames of the files to compress from stdin, for that to work you would have to use:

ls | grep -v gz | xargs gzip

You will exclude files with the pattern gz anywhere in the file name, not just at the end.¹ You also have to take note that parsing the output of ls is dangerous when you have file names with spaces, newlines, etc., are involved.

A more clean solution, not relying on gzip to skip files with a .gz ending is, that also handles non-compressed files in subdirectories:

find .  -type f ! -name "*.gz" -exec gzip {} \;

¹ _{As izkata commented: using .gz alone to improve this, would not work. You would need to use grep -vF .gz or grep -v '\.gz$'. That still leaves the danger of processing ls' output}

Like many commands, md5sum has the ability to read from the standard input if an option's value is - (from man md5sum):

Print or check MD5 (128-bit) checksums. With no FILE, or when FILE is -, read standard input.

Since you know the file name, you could simply print the contents of your md5 file, a couple of spaces and then the name and pass that to md5sum:

$ cat test.txt.md5 
5a6d311c0d8f6d1dd03c1c129061d3b1
$ md5sum -c <(printf "%s  test.txt\n" $(cat test.txt.md5)) 
test.txt: OK

Another option would be to add the file name to your file:

$ sed -i  's/$/  test.txt/' test.txt.md5
$ md5sum -c test.txt.md5 
test.txt: OK