1. I've never heard of a requirement for handrails, irrespective of when the home was built. Actually, it would never even occur to me that there might be such a requirement, especially for three steps. Nonetheless, there might indeed be such a rule. I would call the zoning office. They should at least be able to point you in the right direction.
2. As you know, the definition of a bedroom is very specific. Here's an article that might help:
http://blog.wethmangroup.com/2010/04/09/bedrooms-defined/
I'm having a hard time picturing your specific configuration. It appears that you do not have a large enough basement window, and are relying on the two doors as forms of egress. From my understanding (which you need to verify), the legality of calling the room a bedroom depends on where the doors lead to. At least one must provide access directly to the outside. That is the whole point of the requirement - escape in the event of a fire. So if you have a door leading to the basement, plus another leading to the outside, I believe (but can't confirm) that you're covered.
Having said that, it appears that you are attempting to pass a basement family room off as another bedroom. I'm not sure that that is your best strategy. If you're looking to squeeze the most people into the place, maybe - but that comes with a downside.
Hope that helps.