Can anyone tell me if the Sligo inter-city trains call at Broombridge to connect with the Luas Green Line and if not, then why not? Seems an obvious no brainer.
It does not, and most people would agree with you, it is a no brainer. In fact, you weren't the only one to think of this recently, see this thread here. I believe the OP there has started emailing people about getting the train to stop at Broombridge. PM the OP to find out who they've email, they'll be able to tell you more.
Also, when emailing your local politicians, be sure to include your name and address (this is basically the only type of email in which I'd do this), as they'll always take a query from someone who can vote for them way more seriously.
In the future it seems hard to imagine a sligo train going through glasnevin Station and not stopping.
Give Irish Rail the chance. Every chance they won't.