To add to what Emmett said, also keep in mind that every team name is a registered trademark, as are the phrases National League, American League, NLCS, ALCS, World Series, etc.... so basically in reference to the MLB playoffs if you want to refer to a specific series, you need to say something along the lines of "come see Chicago host LA in first round playoff action" or "the first round of the big league playoffs". Clearchannel used to have a list of all non-allowable terms that their production peeps could refer to, maybe someone has it handy. I do know that Final Four and March Madness were definitely on the list, along with everything even remotely related to the NFL.
And to answer your 2nd question, purchasing one of the broadcast packages for your business allows you to promote the fact that you can watch games there but it does NOT grant you rights to use any of the trademarks in doing so... so again, see the previous Chicago/LA example.
The only exceptions I've seen were NFL sanctioned promotions and spots involving their official sponsors (like Coors or Sprint mentioning the Super Bowl in their spots), but as a rule your sales people are wrong and if they persist you may want to let your GM know that airing the spots with trademarked terms opens the radio station up to a lawsuit.
