Random password cracker using brute force
I'll answer your first question separately. However, since your second and third questions are closely related, I'll give the same answers to both questions.


  Am I following coding standards for Python 2 (like PEP8)


For the most part, your code complies with PEP8,


Spaces between operators
Two newlines between function definitions
Variables are lowercase with underscores as needed
etc...


The only thing I'd suggest is to break some of your longer statements up - such as your print statement at the end of your script - onto separate lines, with each separate line being indented. 

However, even if sometimes you choose not to comply with a certain coding standard, make sure you are consistent with your naming conventions. I've seen Python code which is written Java style, but is still easy to read because the author was consistent in their style.


  Is there any way to make my code more "Pythonic" (like a native Python coder)? and Is there anyway to improve performance, readability, etc.



Instead of having the list_of_chars variable, make use of the string module which already defines all alphanumeric characters:

&gt;&gt;&gt; import string
&gt;&gt;&gt; string.digits + string.ascii_letters 
'0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ'
&gt;&gt;&gt; 

Don't use global variables. There is rarely ever a good reason to use them. Instead of making attempted_password global, define it local to the solve_password() function. This makes much clearer where attempted_password is used.
Inside of your first for-loop, you never use the variable letter. Instead, simply use the variable _ which conveys to the reader that your only using this loop for code repetition:

for _ in range(0, random.randint(100000, 250000)):
    password += list_of_chars[random.randint(0, 61)]

The builtin range() function will already start from zero if no other start value is specified. So there is no need to explicitly do it yourself:

range(random.randint(100000, 250000)) # no need for setting 0