You can find all of those operators in the Python language reference, though you'll have to scroll around a bit to find them all. As other answers have said:

• The ** operator does exponentiation. a ** b is a raised to the b power. The same ** symbol is also used in function argument and calling notations, with a different meaning (passing and receiving arbitrary keyword arguments).
• The ^ operator does a binary xor. a ^ b will return a value with only the bits set in a or in b but not both. This one is simple!
• The % operator is mostly to find the modulus of two integers. a % b returns the remainder after dividing a by b. Unlike the modulus operators in some other programming languages (such as C), in Python a modulus it will have the same sign as b, rather than the same sign as a. The same operator is also used for the "old" style of string formatting, so a % b can return a string if a is a format string and b is a value (or tuple of values) which can be inserted into a.
• The // operator does Python's version of integer division. Python's integer division is not exactly the same as the integer division offered by some other languages (like C), since it rounds towards negative infinity, rather than towards zero. Together with the modulus operator, you can say that a == (a // b)*b + (a % b). In Python 2, floor division is the default behavior when you divide two integers (using the normal division operator /). Since this can be unexpected (especially when you're not picky about what types of numbers you get as arguments to a function), Python 3 has changed to make "true" (floating point) division the norm for division that would be rounded off otherwise, and it will do "floor" division only when explicitly requested. (You can also get the new behavior in Python 2 by putting from __future__ import division at the top of your files. I strongly recommend it!)

+= adds another value with the variable's value and assigns the new value to the variable.

>>> x = 3
>>> x += 2
>>> print x
5

-=, *=, /= does similar for subtraction, multiplication and division.

I don't think it has a name. In the Python Docs under "Unpacking Argument Lists", it's just referred to as "the **-operator."

I'm not sure what you mean by "the other" data structure. When you do f(**kwargs) you unpack the dictionary kwargs as a sequence of key-value pairs. I don't see that there's another structure involved.

I'll copy the example in the above documentation for clarity.

>>> def parrot(voltage, state='a stiff', action='voom'):
...     print "-- This parrot wouldn't", action,
...     print "if you put", voltage, "volts through it.",
...     print "E's", state, "!"
...
>>> d = {"voltage": "four million", "state": "bleedin' demised", "action": "VOOM"}
>>> parrot(**d)
-- This parrot wouldn't VOOM if you put four million volts through it. E's bleedin' demised !

See also: What does *args and **kwargs mean?