There is the make command. It runs the command set in the makeprg option. Use % as a placeholder for the current file name. For example, if you were editing a python script:
:set makeprg=python\ %
Yes, you need to escape the space. After this you can simply run:
:make
If you wish, you can set the autowrite option and it will save automatically before running the makeprg:
:set autowrite
This solves the execute part. Don’t know any way of getting that output into a split window that doesn’t involve redirection to file.