To illustrate an important point, I invite you to take a flight to Egypt next week, buy a spade, and start digging up ancient tombs, without telling anyone that you're going to do so. See what happens to you. It is pretty much the same as what would happen to you in your "digging up an ex-President of the US" scenario. Just randomly popping up in a spot and starting to dig up tombs is always graverobbing, no matter what country you're in.

So most of your questions are addressed by point one below. But the others are important too. It is archaeology under these circumstances:

1. if the investigation is permitted according to the laws of the land;
2. if the investigation is conducted according to binding codes of conduct agreed by professional bodies;
3. if the investigation is conducted in a responsible manner with due attention to professional practices, equipment, and expertise; and
4. if the results of the investigation are recorded, subjected to peer review, and publicly documented in a professional manner.

If point 1 is not observed, it's illegal, unethical, and yes, graverobbing. In most parts of the world this kind of thing will get you put in gaol, your loot confiscated and returned to the authorities in the relevant country, and any hopes of being a respected archaeologist destroyed instantly. Point 2 isn't always an absolute, but it can carry the weight of law under some circumstances (international treaties may still apply even if you're digging up graves where there's no such thing as legal permission, e.g. in a warzone or at the whim of a Somalian warlord). If point 3 isn't observed, it's not archaeology, it's wanton destruction caused by a dilettante doing irreparable damage to irreplaceable data. Both points 3 and 4 will normally be requirements for getting a legal permit in the first place.

There are certainly many more aspects of the professional code of conduct that archaeologists work by, but they're not enforced in quite the same way (e.g. ethical codes regarding the treatment of existing structures and ecosystems, including trees; respect for landowners' rights that may go beyond legal requirements; mutual professional courtesy; etc.). But the above would be a good start.

This doesn't answer your question, but you might benefit by x-posting to /r/AskAnthropology for further discussion and more responses.

When I was working for a private archaeology firm this summer, I know that mostly everything of historic significance was documented, drawn, illustrated, taken photos of, and then repatriated back into the sites. In the US, we have laws in place to protect Native American archaeological digs, including repatriation and even reconciliation of the sites themselves after the digs (Black Mesa, as an example - where after much of the archaeology done by Southern Illinois University, the factory had to reconcile the land after the coal was dug up.)

As was stated, if you ignore the laws, customs, and codes of the government/land, then yes - it is mere grave robbing. If you follow the rules/get the permits/petition for the ability/etc, it becomes ok.

Edit: Typos...

I know there's an archaeology sub, or at least an anthropology sub you could ask this on.

Archaeology becomes grave robbing (not the other way around) when you pay no attention to context, don't record your methodology, don't record how you found things, don't bother checking with local cultural and legal entities (For example : http://en.wikipedia.org/wiki/Kennewick_Man when a native American burial was found in Kennewick, Washington, multiple tribes were contacted to try and establish who actually claimed it, it led to a lengthy legal debate) and generally just want to get things.

The folks who treat archaeology like its Indiana Jones, are grave robbers. It's not a profitable endeavor.

The folks who meticulously record what they found in soil strata by soil strata are archaeologists. Actual archaeology is quite boring in all reality. You spend a month digging about a foot into the ground- sometimes you won't dig anything up at all because there's genuine concern that the attempt will destroy your findings, and this is becoming more and more common practice- write about a book worth of findings, interpretations of evidence, and data, and then publish it, and argue with other archaeologists over whether it amounts to anything.

We do it to better understand their culture and the practices they had of burials back then. Of course, they take whatever treasures they can too so people will go to their museums. I don't think the Egyptians miss it though.

I can say Italian Renaissance figures do get dug up. The Grand Duke and Grand Duchess of Tuscany, Cosimo I de Medici and Eleonora di Toledo, were both removed from their graves, and their burial clothes were extensively examined. If you look through Janet Arnold's Patterns of Fashion (especially the posthumous #4), you'll find plenty of burial clothes and even some blood-soaked shirts with stab holes through them.