Fold the output of grep

Here's what I would like to do: Using gnu grep (or similar one), search files
for some string, STRING, with the output including a context line before and
after each line found. That would look like:

filename:21: This is a context line
filename:22: This line contains STRING
filename:23: This is another context line

I save this output in a file and then open the file with vim. I want vim
to fold the context lines, so I see only lines containing STRING. Now I can
look at the target lines to find the one I am interested in, and if I need
some context to help me, I can unfold the context lines. I should be able to
use gF to open any file and line I want to go to.

For that that would be the perfect grep. I want it in particular, to search
my zillions of saved emails for a message containing a pattern.

It would be nice if I could unfold both the context line before and after the
target line, with one operation. But I do not know how to do that. Is there a
way to make the visible linei of a fold, be the line in the middle of that
fold?

A quickfix list would do some of this, but I do not think such a list can
display output results that include context lines. I tried it once using gnu
grep as the external grep - but it did not seem to work.

Suggestions on how to accomplish this?

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