Don't think sql version is related to ability to 'download files', but 2.4.1+ does supposedly require 5.1.33 of MySQL, so it might be.
Did you look at editing the link to the file that reported 'no input file specified'? Downloading a file isn't really 'downloading a file' any longer. The link clicked up has to check the DB and mdl_files table to find the contenthash named file in /moodledata/filedir/ ... you see a humanly recognizable file name. There is a fairly old discussion concerning Moodles and GoDaddy at:
https://moodle.org/mod/forum/discuss.php?d=36813
Slash arguments?
There is a good suggestion there and you have the tool with 2.4.x:
http://yoursite/admin/tool/health/
That will report problems and attempt to make some suggestions as how to resolve them.
'spirit of sharing', Ken