count number of text or numbers in a cell?

Guys,

I have a bit of a problem.

I need to count the number of character in a cell.

Example: in cell A1 there is 100507, I need a function that will count the characters. IN this case it's 6.

Is there a function that does this?

I have a related question.
Is there a function I can use to count the number of specific characters within a cell?
Using the example below, in cell A1 there is 100507, would there be a way to count how many 0's are in the cell? Could the same thing work with a string of text? If I had ABCDDDEFBCDA, would the same function (if it exists) be able to count how many D's are in the cell?
Also, is there a function that can count how many characters are flanked by the A's?