You could use it...but if the MLB really wanted it they could take it from you. Assuming you were using it as a place for fans to post and chat, I'm sure they wouldn't have a problem with it.
That would definately be a good idea! However even doing so that doesn't grant you the rights to use someone else's trademark. Without permission, there is always the possibility that the trademark holder will take offense at your use and try to take the domain from you through UDRP or lawsuit.
Is it possible you can run your site using MLB in the URL and site title and never hear from the MLB? Of course. It's equally possible the MLB lawyers will file a UDRP against you within the next 5 years.
With MLB, be careful how you use the site. MLB took the historic stance with baseball stats and it's effects on MLB licenced sites/games. There is a lawsuit with CDM Sports against MLB in regards to this.
With that said, anytime you use a TM, there can be trouble. There is legal and reality. Legally, using a TM without permission is against the law, but in reality, the TM holder enforcing thier TM is a different story.
It's highly unlikely they will grant you permission to use the trademarks. At least not for free. This is why a lot of webmasters live by the philosophy "it's easier to ask for forgiveness than it is to ask for permission." If a trademark holder takes offense, often they'll just send you a C&D letter, and you can just turn over the domain name or remove the offending content and the case is closed. It's rare that an actual lawsuit comes out of it.
If you have MLB in the domain name and use the name for a site about baseball, I think you're asking for trouble. If you have MLB in the domain and use the name for a site that's about something different than baseball, I think you'd be alright.