If I am not the skandhas, then what entity/group of entities am I being responsible for by being a good person?

In terms of skandhas, the act of willpower and action is possible due to the volition skandha, particularly a few factors of the 5 omnipresent factors. The 7th consciousness combined with ignorant clinging is also the thing that wraps one up into a self that bobs from life to life, as the 12 nidanas (12 causes) explains.

Death is conditioned by birth . . . birth is conditioned by becoming . . . becoming is conditioned by grasping . . . grasping is conditioned by craving . . . craving is conditioned by feeling . . . feeling is conditioned by contact . . . contact is conditioned by sense . . . sense is conditioned by mind-body. . . mind-body is conditioned by consciousness . . . consciousness is conditioned by forming force . . . forming force is conditioned by ignorance. (SN ii 25)

Even though it is still creating karma, one must build a strong foundation (satipattana, samatha, samadhi, etc.) to stand on can you shine the light (vipasasna, prajna) to stop the process of craving, becoming, and ultimately even the 1st cause... ignorance.

Building the foundation consists of the 2nd training, shining the light is the 3rd training.

The "being a good person" is the 1st training aka making merit, indispensible to gaining entry into the Buddhist realms of cultivation and cutting off the nidanas closer and closer to the root.

In Mahayana who you really are is everyone collectively. (At least my understanding of the Huayen version and my reading of Tom Pepper's work on sunyata)

If I am anatta (not self because composite) then who is the one responsible for his/her actions? Why am I still responsible for my karma?

We are collectively responsible for our actions.

What is it that transmutes from life to life?

This is a logical puzzle, a corner Buddhist painted themselves into. The Buddha had to deal with an overhang of old beliefs surrounding him. Some, like the efficacy of animal sacrifice, he rejected outright. Others, he kept, even though they don't work well with some of the new ideas like annata. Nagarjunas logical gymanatics attempted to address this: http://en.wikipedia.org/wiki/Nagarjuna#Two_truths

Personally, I think enlightenment is the liberation from heaven, hell and the cycle of rebirth because you realize that they can't exist. The consequence is that we are a collective consciousness, which reframes the question of our personal liberation to the question of universal liberation. It also shifts the possibility of immortality from personal immortality (either an unpleasant one involving reincarnation, or a pleasant one involving a neither here nor there state of nirvana) and replaces it with the fact that collectively everyone else lives on after we die, excluding global extinction.

ref: Tom Pepper for a good description of the collective consciousness and alternate interpretations

http://www.nonplusx.com/app/download/708268204/Taking+Anatman+Full+Strength.pdf

He's very lucid & doesn't resort to "well you just can't understand" or "faith" as explanations.

Let's say you knock your head on the wall and feel pain.

I'm guessing that in future, you will avoid knocking your head on the wall. Is it because you think you have a self? Is it because of some sense of responsibility? Is it because you think you have a choice?

No. You will avoid knocking your head on the wall in future, because you want to avoid feeling pain again. It's very practical. So, the question to ask oneself with regards to karma is "what should I do or avoid doing in future, that will lead to my long term happiness?"

Also, it is not useful to analyse whether there is a self or there is no self. Instead, it is more useful to ask, "how does the perception of the self arise and cease?", "how does the perception of pain arise and cease?", "how does the will to make decisions and volitional actions arise and cease?"

Would thinking or believing that "all phenomena is not self", prevent me from experiencing pain in future? No. This is because the perception of self is relatively true and equally true as the perception of pain.

Since the objective is to end suffering, it is better to ask questions that are more conducive for leading towards this goal.

Rebirth is a more difficult topic that is answered here.

I'll answer in two parts.

Regarding not self: The body, feelings, perceptions, volitions and consciousness exist. They just don't exist the way you think they do. We have a misconception about these aggregates and the misconception is that there is an entity, the "I", that is somehow the essence of these aggregates. This "essence" is conceived as permanent, independent and uniquely individual. It is the "me", or "I". The deluded enchantment with this entity, and with what can bring happiness to it and what can bring suffering to it, is what causes suffering. Our whole life is trying to be happy and avoid suffering based on this misconception. But this misconceived entity does not exist. There is nothing permanent and independent about the aggregates. The aggregates exist, but are in a constant flux. And when this is seen clearly, there is no permanent and independent entity found. This is such a radical change in perspective that our whole life changes when we fully realise it. There is no more suffering.

Regarding choice: It is true that everything arises and falls due to causes. It seems like the inevitable logical consequence is that we are but puppets in this everchanging and causal universe. In a way that's true. Let me give you an analogy: say I throw you a ball for you to catch. You know that it is in the nature of the ball to be subject to gravity. But how does that help you to catch the ball perfectly? It is of little help, really. In the same way, you can observe that these aggregates are constantly changing due to causes, so they are completely subject to causes, just like a ball is subject to gravity. But that doesn't mean that you know precisely how your actions will affect others, just like you don't know the precise trajectory of the ball. So when you don't know with full precision what are the specific consequences of your actions, you have several outcomes and, thus, a subjective choice. That's why it's best to choose to act ethically. To act with kind intentions usually brings the best results for you and others.

***Very simply, the illusory self. This is the director of one's actions. This is the orchestrator of our fantastic illusory dramas life-time after life-time. You are responsible for your actions because you act on behalf of this illusory self and accumulate karmic traces and patterns in your mind stream because of your attachment to this illusory self. On the basis of our extremely subtle luminous natures, the karmic winds transmute and propel you to your next life. It is possible to do good and bad because we are very much caught in the relative manifestations that occur, despite the fact that they are illusory and that nothing - absolutely nothing exists as it appears. However, the lighter the karma becomes the easier it is to blend with our ultimate true nature. Eventually, we get to a stage, after a great deal of purification, accumulation of merit and meditation, where there is no distinction between good or bad - Everything is "All Good". That is the Kingdom of Samantabhadra.

Why does a seed grow into a plant and then to a tree? Why not just stay as a seed? Does it require a soul for that or does it require fertilizer, sunlight, air, a viable seed etc. Does water require a soul to turn into snow in the winter season? Regarding Karma, why does the seed of a mango tree gives a mango plant, but not a coconut plant? Why does a plant of a mango tree have qualities similar to the parent tree? Does it require a soul for that? Certainly not!

"Then, in the mind of a certain bhikkhu this thought arose: “So, it seems, material form is not self, feeling is not self, perception is not self, formations are not self, consciousness is not self. What self, then, will actions done by the not-self affect?"

The same question was raised by a bhikkhu in MN 109, and the Buddha provided a great answer here Sutta's synopsis:

"A thorough discussion of issues related to the five aggregates. Toward the end of the discussion, a monk thinks that he has found a loophole in the teaching. The way the Buddha handles this incident shows the proper use of the teachings on the aggregates: not as a metaphysical theory, but as a tool for questioning clinging and so gaining release."

Buddhist concept of not self is always difficult to translate. It generally means:

• There is no core or soul
• You have no absolute control over your aggregates or the world at large

Also when you have the perception of self you have a yard stick which you use to measure other people, their actions, standing, wealth, etc. with respect to you. This creates misery. If you drop the yard stick then you get out of misery.

Having said this though there is no lasting entity as you there is a mind matter process. What ever you do becomes fabrications which at that point influence the process and also has a "loop back" at different times.