Yes. Atheist simply means you lack belief in a god. It says nothing about your belief (or lack thereof) in fairies, ghosts, leprechauns, and anything else. As long as it isn't a deity, you can be an atheist and believe in it.

I can't figure out what this would be, so I'm saying "no".

I don't believe one can be agnostic and athiest as the definitions conflict. An athiest does not believe in the existence of any god, as an agnostic has neither belief nor disbelief. If you twist one or the other it's close, but most agnostics I've met don't believe, but they don't disbelieve, they just don't have evidence one way or the other. TLDR an athiest KNOWS there is no god, an agnostic doesn't KNOW either way. Personally I would consider myself an agnostic according to definition, but that's only because I can't KNOW there is no god(des/s) without some sort of evidence, but I have myself marked down with the athiests, because I am pretty sure if evidence (if ever found) will conclude that there is indeed no higher entity.

As far as spiritual goes... that just means you believe that there is a human soul, that there is something else besides the physical realm. That much I do believe. I don't believe our consciousness is delivered to some paradise or damnation, only that there are things that can't be explained, and that may just be a 'yet' but I do believe as we evolve as a species our minds are more than just an electrical system for our bodies. It may just be superstition on my part, but I've had too many unexplained coincidences in my life to feel comfortable saying I don't believe that people have souls... I just don't know what happens when the lights go out.

I don't know how to juggle.

I think you'd have to define all 3 terms (in your terms!) for us to be able to answer.

The "theist" part of Atheist refers to deity. Agnostic just means god doesn't seem probable but there is still a possibility. "Spiritual" , as cosmicrunch points out, can mean pretty much anything you want. I think it typically means there is a spirit realm that exists beyond the capacity of our 5 senses to identify.

Yes. In using our sixth sense, empathy, to understand others, is about as spiritual as humans can get.

Personally I avoid the term 'spiritual' simply due to the broad way it may be interpreted to mean. Far too many apply it to the supernatural for me to be comfortable using it. However in my head I can agree with a lot of the way Sam Harris describes it.

Given the fact that the word "spiritual" can mean literally whatever you want, go for it.

Yup. I think it's like being ambidextrous.

Sure. You could believe that life evolved, without a giant rabbit that made the universe, but people and animals are connected in a spiritual network and/or can exist beyond death.

Read Waking Up by Sam Harris, it talks a lot about spirituality without the funny stuff. I myself couldn't get into it but he's a respected atheist exploring rational spirituality.

Why not? I have a friend who is very spiritual, talking about the potential goodness in all of us and the healing power of meditation and the beauty of nature. There is nothing religious or cult-like about her message.