Finding exact match in vim

Question 1

To find a single, standalone aa (not a, aaa, ...), you need to assert no match of that character both before and afterwards. This is done with negative lookbehind (\@<! in Vim's regular expression syntax) and lookahead (\@!) enclosing the match itself (a\{2}):

/\%(a\)\@<!a\{2}\%(a\)\@!/

Simplification

Those assertions are hard to type, if the border around the match is also a non-keyword / keyword border, you can use the shorter \<\a\{2}\> assertions (as in LSchueler's answer), but this doesn't work in the general case, e.g. with xaax.

Question 2

You enclose the string you are looking for by \< and \> like in /\<aa\> to match exactly that string.

Question 3

You can search for aa[^a], which will find your two as and nothing else.

EDIT: oh boy, y'all are after an exact match :-) So, to match exactly two aas and nothing else:

[^a]\zsaa\ze[^a]\|^\zsaa\ze$\|^\zsaa\ze[^a]\|[^a]\zsaa\ze$

And the branches expanded out:

  [^a]\zsaa\ze[^a]
\|^\zsaa\ze$
\|^\zsaa\ze[^a]
\|[^a]\zsaa\ze$

These cover all contingencies--the aas can be at the beginning, the middle, or the end of any line. And there can't be more than two as together.

\zs means the actual match starts here
\ze means the actual match ends here
the first branch finds aa in a line surrounded by other characters
the second branch finds aa when it makes up the whole line
the third branch finds aa at the beginning of a line
and the fourth branch finds aa at the end of a line.

My mind boggles at fancy things like look-behind assertions, so I tried to stick to reasonably simple regex concepts.

EDIT 2: see @benjifisher's simplified, more elegant version below for your intellectual pleasure.

Question 4

This simple one helped me to match the exact pattern. You can try this.

\V\c\<pattern_here\>

Question 5

Bring your cursor to exact word which you want to search.

Press * (Either Shitf+8 or from numpad of keybord).