A tricky topic! We have 5 children, so we deal with hurt feelings/conflict on a regular basis. It is very important to DH and I to help our children get along, to maintain peace in our family. When there is an offense/conflict/whatever, we try to encourage both parties to share what happened, validate the feelings, and to give them time to calm down. We try to get to the root of the problem, talk about how they could have handled it better, what they would like to have done differently, etc. When the child is calmer, we do remind them that the long-term relationship is more important than the current situation, and that they really do want to preserve it. We don't force apologies, but we do encourage them to try to find a way to mend the relationship (which doesn't necessarily happen instantaneously, lol). Sometimes, it's very helpful to let one or both children sit by themselves to calm down and figure out what to do. Sometimes, they really need our help to get past the anger/hurt and figure out how to approach the other child. One thing that we have found helpful is to remind the child that they can rejoin the fun when they have apologized and done their part to fix the problem (because they are not always ready to move forward at the same time).
I don't think that forced apologies do much for either party; one feels resentful for being forced to apologize, the other feels lied to. It does nothing to restore the relationship, increase the empathy, and teach children how to effectively deal with conflict, which, IMO, is the goal. I know that our approach isn't always successful, but most of the time, our children do end up apologizing to each other. Most of the time, once they get past the initial emotion, they are genuinely sad that they caused hurt to one of their siblings, and do want to make it right. I think that it is our job as parents to help them figure out how to do that.
I do like the PP where the child drew a picture and dictated the apology for breaking apart the other child's lego structure. I've used drawing/art in that manner before quite successfully, and have found that it is a good medium for expressing feelings on the part of both children. It is very non-confrontational, which IME makes it easier for children to apologize, and to accept the apology.
RE: the adult/child issue. When one of my children does something particularly rude or disrespectful to me, I let them know how I feel (hurt, disappointed, offended, whatever), and that I am sad when they act that way. I do prompt an apology by telling them that it would help me feel better, just like it helps them to feel better when they receive a sincere apology after an offense. Usually, when they understand the effects of their actions, they are sorry and ready to apologize (not always, lol!). They will usually apologize and give a hug. However, again, I don't think that a forced apology really does much to help the situation. I mean, as the adult, I'm already bigger and stronger than most of my children, how in the world does forcing them to apologize make them feel any better about being smaller and weaker than me (which is often the reason they do something disrespectful, they are trying to be more powerful). Usually, just letting them know how I feel after something like that is enough for them to want to apologize, just like it is for me.