Why is Sage avoiding her father?
Sage actually explains it to us. She wants to stand on her own two feet, and as we saw Stephen and Jade at the party, they don't think much of talking, instead buying their children's love with their money. Tybalt gladly accepts the offer, but Sage doesn't. You can't buy a child's love with money. Jill said something similar to the MC when he stayed over in EP3.
Sage actually has a hard time asking her parents to help Maya. From her point of view, it is even understandable because she is indirectly admitting to her parents that she cannot do some things without their help. For Sage, it's a defeat.
I agree with you on the latter. It could be that Sage knew Stephen was cheating on her mother Jade with Cathy. Sage talks to Quinn about it, and she makes a plan to get rid of Cathy. (In EP8, Quinn wants to take care of the Alphas.) This was before the MC came to B&R. Quinn involves Mel and Sarah in her plan. One of them takes the picture of Cathy in the gym and the other stands lookout. The girl watching happens to make another discovery. She sees, as the MC later, Chad and Troy kissing and talking. Chad then gets a pair of panties put in his gym bag. Why this. Josy explained it to us in EP8. It would be worse for Sage if Chad was gay and not bisexual. Sage finds the panties and naturally suspects Chad is cheating on her. But because Melanie and Sarah found out that Chad and Troy love each other, they want to help them. That's why Sarah tells Mel in EP8 that the truth would be much worse for Sage.