a mutable type inside an immutable container

You can always modify a mutable value inside a tuple. The puzzling behavior you see with

>>> thing[0] += 'd'

is caused by +=. The += operator does in-place addition but also an assignment — the in-place addition works just file, but the assignment fails since the tuple is immutable. Thinking of it like

>>> thing[0] = thing[0] + 'd'

explains this better. We can use the dis module from the standard library to look at the bytecode generated from both expressions. With += we get an INPLACE_ADD bytecode:

>>> def f(some_list):
...     some_list += ["foo"]
... 
>>> dis.dis(f)
  2           0 LOAD_FAST                0 (some_list)
              3 LOAD_CONST               1 ('foo')
              6 BUILD_LIST               1
              9 INPLACE_ADD         
             10 STORE_FAST               0 (some_list)
             13 LOAD_CONST               0 (None)
             16 RETURN_VALUE        

With + we get a BINARY_ADD:

>>> def g(some_list):
...     some_list = some_list + ["foo"]
>>> dis.dis(g)
  2           0 LOAD_FAST                0 (some_list)
              3 LOAD_CONST               1 ('foo')
              6 BUILD_LIST               1
              9 BINARY_ADD          
             10 STORE_FAST               0 (some_list)
             13 LOAD_CONST               0 (None)
             16 RETURN_VALUE        

Notice that we get a STORE_FAST in both places. This is the bytecode that fails when you try to store back into a tuple — the INPLACE_ADD that comes just before works fine.

This explains why the "Doesn't work, and works" case leaves the modified list behind: the tuple already has a reference to the list:

>>> id(thing[0])
3074072428L

The list is then modified by the INPLACE_ADD and the STORE_FAST fails:

>>> thing[0] += 'd'
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
TypeError: 'tuple' object does not support item assignment

So the tuple still has a reference to the same list, but the list has been modified in-place:

>>> id(thing[0])
3074072428L
>>> thing[0] 
['b', 'c', 'd']

You can't modify the tuple, but you can modify the contents of things contained within the tuple. Lists (along with sets, dicts, and objects) are a reference type and thus the "thing" in the tuple is just a reference - the actual list is a mutable object which is pointed to by that reference and can be modified without changing the reference itself.

( + ,)       <--- your tuple (this can't be changed)
  |
  |
  v
 ['a']       <--- the list object your tuple references (this can be changed)

---

After thing[0][0] = 'b':

( + ,)       <--- notice how the contents of this are still the same
  |
  |
  v
 ['b']       <--- but the contents of this have changed

---

After thing[0].append('c'):

( + ,)       <--- notice how this is still the same
  |
  |
  v
 ['b','c']   <--- but this has changed again

---

The reason why += errors is that it's not completely equivalent to .append() - it actually does an addition and then an assignment (and the assignment fails), rather than merely appending in-place.