Incompatible changes to the signature of a method. This doesn't happen as much in Python as it does in Java and C++.
Python has optional arguments, default values, and far more flexibility in defining method signatures. Also, duck typing means that -- for example -- you don't have to switch from some class to an interface as part of a significant software change. Things just aren't as complex.
How do you find all the places where that method is being called? grep works for dynamic languages. If you need to know every place a method is used, grep (or equivalent IDE-supported search) works great.
How do you find out what operations an instance provides, since you don't have a static type to lookup?
a. Look at the source. You don't have the Java/C++ problem of object libraries and jar files to contend with. You don't need all the elaborate aids and tools that those languages require.
b. An IDE can provide signature information under many common circumstances. You can, easily, defeat your IDE's reasoning powers. When that happens, you should probably review what you're doing to be sure it makes sense. If your IDE can't reason out your type information, perhaps it's too dynamic.
c. In Python, you often work through the interactive interpreter. Unlike Java and C++, you can explore your instances directly and interactively. You don't need a sophisticated IDE.
Example:
>>> x= SomeClass()
>>> dir(x)
How do you handle/prevent typing errors? Same as static languages: you don't prevent them. You find and correct them. Java can only find a certain class of typos. If you have two similar class or variable names, you can wind up in deep trouble, even with static type checking.
Example:
class MyClass { }
class MyClassx extends MyClass { }
A typo with these two class names can cause havoc. ["But I wouldn't put myself in that position with Java," folks say. Agreed. I wouldn't put myself in that position with Python, either; you make classes that are profoundly different, and will fail early if they're misused.]
Are UnitTest's used as a substitute for static type checking? Here's the other Point of view: static type checking is a substitute for clear, simple design.
I've worked with programmers who weren't sure why an application worked. They couldn't figure out why things didn't compile; the didn't know the difference between abstract superclass and interface, and the couldn't figure out why a change in place makes a bunch of other modules in a separate JAR file crash. The static type checking gave them false confidence in a flawed design.
Dynamic languages allow programs to be simple. Simplicity is a substitute for static type checking. Clarity is a substitute for static type checking.