This game is more commonly known as Lights Out, and has a number of elegant solutions, all based in some standard but somewhat advanced mathematics. I won't describe them all here but if you Google a bit you can find all kinds of explanations varying from straightforward procedures to transformations into linear algebra or group theory. A few links:
http://www.hamusutaa.com/pilot/solution.html
http://www.ripon.edu/academics/macs/summation/2010/articles/M.%20Madsen%20-%20Lights%20Out.pdf
http://people.math.sfu.ca/~jtmulhol/math302/notes/24-Lights-Out.pdf
Edit: Re: your second question. The algorithm presented in the second link I posted can solve an n x n board in O(n^6) time, meaning you should be able to quickly solve a 25 x 25 board.
