If I run a script, is there a way to run it again without doing
You can create an action, assign an F-key to the action, then use the F-key.
Insert Menu Item command located in the Action Panel Menu to insert the call to the script.
But be aware, when you quit Illustrator, the action loses the connection to the script. So, each time you launch Illustrator, you have reconfigure the action. This is a long standing bug in Illustrator.
Unfortunately, the answer is no. This has been a requested feature for years now and Adobe has done nothing to address it.
Ctrl + F12 gets you into the Open Scripts dialog (in Windows, anyways)...It's not an answer to your question but I sometimes find it to be quicker than clicking File and going to Scripts.
In Windows I use AutoHotkey for things like this.
To use F1 to run File > Scripts > Script Name it would look like this...
Edit: I forgot to mention if you want this shortcut to be Illustrator specific so it doesn't affect any other programs just add the following...
#IfWinActive, ahk_exe Illustrator.exe F1::Send !frs #IfWinActive
I think Ctrl/Cmd＋D can do it. I practiced it with general actions in AI and other scripts.