Yes, "на основании" fits here.

Сообщение = message, but yes, in news-oriented media, it could mean: Yeah, we (the journalists) have just received the message and we are passing it to you. This message will be probably a new information to you, our audience, so for you it is Новость.

на новостях - difficult to explain. I'd say it's prepositional and it is being applied because the question is "на чём"
It's like "Принцесса [лежит] на горошине", "Цветок [стоит] на подоконнике", "Кинотеатр находится на площади" etc. I understand, that "на чём? - на новостях" can be not so obvious as for native speaker, but still "на чём?" is correct question here.
On the other hand, на is used also in accusative (question куда): "Положи принцессу на горошину", "Поставь цветок на подоконник".
На cannot be used with Dative, never.