I doubt it, Hashi's dad thinks war was the only way for peace rather than Hashi and Madara think of finding peace with compromises which is why they get along. Butsama just doesn't seem like he has the attitude to have a "friendly" rivalry. You could say their rivals because they're both leaders of separate clans though.
Nope, they were in an era of clan vs clan. They were nothing more than enemies on the battlefield, being the leaders of their clans they obviously sought to outdo each other, but they were never on friendly terms at all. They inherited the same hatred from their own fathers, the cycle continues down to Madara and Hashirama.
No, they obviously know about each other because this is a war and to be successful you need to know who and what you're up against.. no surprise that the Uchiha know the Senju members and vice versa