Next To Something - Grid Game

Here is a simple game. (Possibly very unoriginal.)
(Do not worry, there is no more math than in Tic-Tac-Toe, and the game
could possibly be understood and enjoyed by children too.)
You start with a grid which is the same number of squares high as
wide.
(I suggest a 5-by-5 grid for beginners.)
The game is for 2 players, each with a different colored pencil
(or one player uses a regular pencil, the other uses a pen).
Players take turns filling in the yet-unfilled squares, one square per
move, as follows:
Player-1 can fill in any yet-unfilled square along the grid's _bottom_
or
_top_ borders, or he/she can fill in any yet-unfilled square
immediately
_above_ or _below_ a square already filled in by her/his opponent,
player-2.
Player-2 can fill in any yet-unfilled square along the grid's _left_
or
_right_ borders, or she/he can fill in any yet-unfilled square
immediately _left of_ or _right of_ a square already filled in by
his/her
opponent, player-1.
The winner is the last player who fills in a square.
Notes (mostly unnecessary):
Players can use the same color pencil/pen too, but one player fills in
the squares with x's, the other fills in the squares with o's (or
with
whatever symbols the players choose).
The corner squares can be filled in by either player.
If there is an even number of squares, player-2 might try to match
player-1's moves (but rotated by 90-degrees), so as to fill in the
grid
completely and be the last to fill in a square. This strategy can be
easily countered, however, by player-1.
I said the winner is the last player who fills in a square, instead of
saying the winner is the last player who is ABLE to fill in a square,
because part of the game is not overlooking any possible moves a
player
can make.
(This is much more of an issue with larger grids.)
In any case, the player who (thinks he/she) cannot move concedes to
the
other player at game's end.
Have fun!
Questions:
1) What are some strategies? Anything certain to win?
2) What is the number of the most pairs of squares filled in by the
same player which can, in theory, be immediately next to each other in
an n-by-n grid?
(Zero is the least number of same-player squares that can possibly be
adjacent. {Squares filled in with checkerboard pattern.})
Thanks,
Leroy Quet