Surreal has the definition that you give, but it typically doesn't describe a person or object, but instead an experience or feeling. You wouldn't normally describe someone or something as habitually surreal, even if they are strange and unusual. Your example 1 works, 2 less so, and 3 not really, the way it is written. I've rewritten 2 and 3 with a bit more context to clarify how you might use the word in those situations.
Surreal is neither negative nor positive. It can easily go either way. It is most often used for an experience that is truly otherworldly (deja vu, perhaps, or the sensation of time slowing down when you are in danger) but is occasionally used sarcastically to describe completely mundane things that are mildly unusual (not losing a sock in the laundry, walking across a dark room without stubbing your toe, that kind of thing.)
- John went crazy when he met Justin Bieber. Watching a normally quiet 40 year old man turn into a 12 year old girl was surreal. (Here John isn't surreal, his actions are surreal.)
- Watching their dog interact with strangers is surreal. It acts wild but never barks.
(Not the best usage, but the best I could think of with a dog-surreal combination. Strange, weird, bizarre, or unusual would be preferable here, in my opinion.)
More examples:
4. When I realized that our cars were about to crash, time slowed down; it was surreal.
5. I am not a handsome man, so it felt surreal when the most beautiful woman at the party asked me to dance.
6. My drive to work was surreal, every light was green along the way and there was no traffic at all. I hardly even saw another car.
Here is a page with many more sentences using the word as well. Some are better than others, but all of them are sensible.
I think you understand things correctly.
How is used to ask about the quality/degree of something, or the manner in which something was done. When you ask someone how is X, they may respond with a details in a similar manner as what, but the reason is to communicate a quality/degree - typically "good" or "bad."
So...
How is your teacher?
She's this really weird tall lady with these crazy glasses, and she was yelling at everyone today.
The implication here is that she considers the teacher a bit intimidating, so somewhat "bad." And that's what the person asking how really wanted to know.
What's X like to me, implies that you are less interested in the person's personal feelings - rather, you want to know how X compares to other X's - than how's does. But this can be influenced either way greatly by context.
Q: What's the weather like over there?
A: It snows a lot down here, and sometimes it rains.
Q: So usually, how's the weather over there?
A: It snows a lot down here. But it's raining now.
Q: How's the weather?
A: Rainy. Take a jacket.
Q: What's the weather like?
A: Today? It's rainy.
Best Answer
The difference is a matter of emphasis.
The first, with the present perfect
presents her recent history as involving a number of discrete events.
The second, with the present perfect continuous
presents her recent experience as a continuum of strange events: things are happening to her seemingly one after the other with little or no opportunity to recover from the event or to assimilate it before yet another strange thing comes along. She has been "beset" with strange events.