In general, there is little or no difference between them; the three words can be used almost interchangably in many situations (particularly anything and whatever). But your examples manage to point out some subtle contextual differences.
Do whatever you like.
Do anything you like.
These are pretty much synonomous. Figure out something you want to do, and do it.
Do everything you like.
This is more all-encompassing. Instead of focusing on one thing to do, you can do several things.
Perhaps it's better described with a more concrete example. Suppose a parent and child are brainstorming birthday ideas, and they've come up with four: have an afternoon party, meet with friends at a restaurant for dinner, go skating, or go bowling.
If the parent says:
For your birthday, we can do whatever you'd like.
The implication is: pick one (or maybe two) of those options, and we'll do it.
But if the parent says:
For your birthday, we can do everything you'd like.
The implication is: it may be an exhausting day, but we can do any or all of those things!
Whatever you do, do your best.
With anything you do, do your best
In everything you do, do your best.
All three of these are ways to say, "Put forth your best effort." The third one seems more universal, but the end result is the same. It doesn't matter of you do your best every time you start a new task (Whatever you do, do your best), or if you live by the exhortation to do your best at all times (In everything you do, do your best). No matter how it's worded, live by this principle, and you'll always be doing your best.
Stop doing whatever I do!
Stop doing everything I do!
Both of these sound like someone is being chided for aping someone else. But
Stop doing anything I do!
doesn't sound quite right. Anything seems too vague to refer to some explicitly mimicked action, but whatever and everything work just fine.