Yes, that is a rooster, and yes the hen will have a high chance of surviving.
Also size is not what wins a fight, the two roosters may live together with very few fights, age and breed are the main factors that determine the outcome of a fight. I have 4 mature roosters in my flock and the oldest is the king, he is around 4 and is a RIR, his spurs are over two inches in length and the other roosters do not challenge him, ever. The next in charge is a 2 year old Ameraucana he is the smallest of all my roosters but has inch long curved spurs with a sharp tip, he is never challenged by the other two roosters and he never challenges the King. The third in charge is a massive Black Australorp, he is around a year old, he is over twice the size of the Ameraucana but does not ever challenge him because his spurs are only around half an inch long and still have a blunt end to them. And at the bottom of the rooster pecking order is a little white leghorn male, he is 6 1/2 months old and he is a little bigger than my Ameraucana, his spurs are just coming in and he won't be fighting for his place for a while. My roosters don't fight because they know who they can beat and who they can be beaten by, they avoid stronger roosters and the weaker roosters avoid them. Fighting only gets bad between equally matched roosters, roosters of the same age and breed will constantly be challenging each other. Roosters of different breeds and ages make a pretty nice pecking order, like the hens, because they are not equally matched, there spur size is different, there size is different, and there reflexes are different. So hopefully your rooster can beat up the red rooster and create a pecking order, if they are equally matched then you will have to get that red rooster out of your flock. Also if your roosters do create a nice pecking order, make sure you have enough hens for them both. You can also have many rooster when you have large breeding flocks but that's not at all relevant in your situation.
