I like these two together and would very much enjoy seeing more of their friendship, yes. Leliana's behavior and facial expressions during the conversation about how to solve Josephine's personal quest are quite telling, too: concern, affection, bemused exasperation with Josie's determination to solve things the "right" i.e. diplomatic way, but also respect for the fact that her friend does have the right to do things her way. It was touching. I agree with whoever called Josie a positive influence on Leliana's conscience, as a counter to Marjolaine's callous evil.
I agree, and it's one of the reasons I love their relationship so much (it's probably my favorite in the whole game). Josephine's codex entry is basically Leliana asking Cassandra to appoint her as ambassador for the Inquisition: she proceeds to list all her qualifications and then concludes with a more personal note, something like "if that's not enough, please consider she is also a trusted friend". I think at that point Leli needed a familiar, balancing presence in her life, someone whose "almost painful integrity" would definitely clash with her recent methods and force her to confront the issues she was already having long before the Conclave; someone who rejected the life Leliana had to fully embrace, and yet didn't judge her for it. She likes to tease Josie, to make fun of her constant worrying and looking for "needlessly" peaceful solutions but she's perfectly aware that the Inquisition needs that perspective as well. And Josephine may not be directly responsible for Leliana's (un)hardening but I love how the playful banter, the pranks, the tea parties (how upset does she get when she hears Josie is disappointed?), the glimpses of a lighter, "softer" Leliana we get when she's around her are always there, regardless of the path we choose for her.