The first one looks like a team hand stack (warning: black hole ahem, sorry, TV Tropes link). It usually involves more than two people, though. (Perhaps that gets hard to photograph?)
The second is an arm wrestling handshake, since that is the usual position of the hands when arm wrestling.
The third looks like both people are doing a double-hander - see the description a bit past halfway down that page. Alternatively, I've seen this called the glove handshake. This type of handshake is a stereotypical favorite of politicians.
I'm not coming up with a name for the fourth. It looks kind of like a variant of the double-handed handshake (see above), just more sincere. If I had to describe the scene, I'd probably use some variant of "clasp":
She clasped both of his hands between hers and gave him a warm smile.
Edit: Since you asked especially about the first picture...
Like I commented, the first picture does not read "greeting" to me, nor —I dare say— to anyone else in the Western world. Thus, if you wanted to talk about a greeting that involves stacking hands like this, you would have to describe it.
We greeted each other in the Akha manner, clasping all four of our hands in an alternating stack and scrunching up our eyes.
or
We greeted each other in the Akha manner, piling our hands on top of each others' as if we were starting a team hand stack, and scrunching up our eyes.
Naturally, once you've defined/described the greeting, you could refer to it as an "Akha handshake" in the rest of your narrative.
"What's up" when used as a greeting follows the pattern of How do you do. It may appear at first glance appears to be a question, but it is in fact stated and given as an idiomatic set phrase, the correct response to which is to repeat the question:
John: Hey Matt! What's up?
Matt: Yo John! What's up? How've you been? I've not seen you since Monday!
"What's up" is an informal greeting, and should only be used in informal situations. If in doubt, do not use it to greet people that have not greeted you with "What's up" in the past. For more formal situations, use hello or hi.
Best Answer
I expect this kind of greeting is a cultural artifact. In most of the English-speaking world, we ask how you are, not whether you have eaten.
Americans (and, I assume, the British and other English-speaking nationalities) do not casually ask if someone has eaten, as it is normally considered an invitation, and usually includes some offer of food or drink. For example, suppose John has come to visit Alan, who he doesn't know very well. Alan wants to seem hospitable, so he offers John some food:
Side note: In a similar way to the way your culture asks "Have you eaten", in the English-speaking world "How are you?" is rarely an invitation to talk about your actual condition. It's just a polite salutation. The expected response is, "I'm good, how are you?" even if you are not well.