Is it possible to run Python code from within the vim editor?
What is necessary to install the support along with Python syntax highlighting?
How would I install "python.vim : Enhanced version of the python syntax highlighting script" ? 
I did not automatically create ~/.vim/syntax and I'm using a Mac, all I downloaded was the .app file, an executable that I don't know of its purpose and a readme file.
I've tried also creating a folder for the python.vim file, but that didn't work out either.
Personally:

When inside Vim editing my Python scripts, I simply hit CtrlZ so as to return in console mode.
Run my script with command $ python my_script.py.
When done, I enter $ fg in the command line and that gets me back inside Vim, in the state I was before hitting CtrlZ. (fg as in foreground)


Edit
Recently I have started using the :terminal mode of vim much more frequently.
I tend to prefer it to CtrlZZ because it may happen that I forget that I used Ctrl-z and open an additional vim session: it may become messy. Also, having a terminal pane is easier for dealing with line number in errors message, since the two views are available at the same time.
So the workflow I'm using nowadays has become:

:terminal  (in my case I have a vim mapping with leader key) <leader>tm :terminal<cr> so that I don't even type :terminal manually)
Run my script with command $ python my_script.py.
$ exit in the bash command line if I want to close the terminal pane
Can Python be run from within the vim editor?