Given that Lily was at least 20 years younger than Robert, I imagine that had he not left her after the crash, he might have been the one to take responsibility for her. He would have comforted her over the loss of her parents, and again when the truth about her parents--about Lily being broke was discovered. Robert might have reorganized the finances so they could keep the house; Robert might have sorted the bills; and Robert might have packed the boxes if they did have to move after all. But he wasn't there. Lily grew up a lot during this time. She was alone and learned to rely on herself. She learned how strong she was.
I think while Robert was working and busy having his affair with Rita, Lily was lonely. She missed talking to her husband, listening to music, and drinking a beer on the porch with him. So she really enjoyed her time with Piedmont. She found him interesting, talented, and intelligent. She had been without a gentle touch and comfort for her losses- -Piedmont filled a hole in her heart. A little part of Lily loved him for that. I also think she always knew Robert would come back, and she would welcome him home- -so she was confused about how she could care so much for Piedmont.
Rita knew ultimately that she could not stay with Robert because he was married. Lily knew that Piedmont could not be the one to mend her heartbreak because she was waiting for her husband to return. She related more to Rita- -understood why she continued to see a married man- -you can't stop your heart from feeling what it is feeling even if it isn't the best thing or the right thing for you. The heart can surprise you! I don't think Lily felt sympathy for Rita necessary because she died, but because she understood about more about where Rita's heart was.
Lily wanted to continue her friendship with Piedmont and include Robert in it. Piedmont was not interested in being friends, probably more out of fear than anything. He was terrified when he heard the door jiggle the night Lily went into labor. given the time, if the two were friends, people would assume the worst about Piedmont's role in their relationship and maybe punishPiedmont like they did Emmett Till.
Lily also understood a bit more ore about Robert's heart after his time away. She was gentle with him and gave him time and space to grieve for Rita.