Buzzword Bingo Boss Battle!

So, we are playing misère Tic-tac-toe with the option of taking more than one turn in a row. Seems quite interesting, so let's take a stab at it.

The flow of the narrative here is a bit messy, because the answer was not all written in one go. Since a full rewrite would be too time consuming, I've added some chapter breaks to give at least some structure to the text. Here's what's going to happen:

1. Getting acquainted with the game, figuring out the best starting move
2. Break down what we learned as explicit heuristics
3. Analyse a particularly likely-seeming 4-move position
4. Using that result, MiniMax our way back to the first move
5. The Conclusion

So, here we go.

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Getting acquainted with the game, figuring out the best starting move

At first glance, it seems that the centre must be avoided at all costs. Indeed, if you start at the centre, the opponent has a "mate-in-2":

(The starting player can play a second 3 at one of the 4s himself, but since he cannot take both, the result is still the same.)

Similarly, playing in a corner seems questionable:

Playing move 3 at either triangle instantly loses to the opponent leaving only the other triangle, and picking any other spot creates another unplayable square, which the opponent can leverage for an eventual win:

Taking any additional squares in combination with these bad squares is only going to make things worse, so the best starting move must be to

This agrees with the intuitive notion that you want your plays to create as few potentially losing connections as possible, and the sides are the only squares that are not on a diagonal.

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Break down what we learned as explicit heuristics

From here the variations branch a bit, so let's establish some important heuristics first:

1. In general, sides are better than corners, which are better than the centre. This is because of the number of possible losing lines going through the squares.
2. Making an "open 2" (so that there's an empty square that only the opponent can play) is generally bad
   2.1. Conversely, forcing the opponent to make an "open 2" is good
3. Blocking is mutual: if you block some of your opponent's lines, some of your lines are already blocked.
   3.1. Aggressive moves often leave a counter-attack possibility
   3.2. It's easier to defend than it is to attack.
4. Playing several squares at once is generally bad unless you have a definite plan in mind.

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Analyse a particularly likely-seeming 4-move position

In light of these, let's examine the most likely opening:

(If 3@4, then 4@3 wins, so this is the only plausible 4-sides opening.)

Looks like playing the 5th move in the "most blocked" corner gives a draw here:

(Somewhat surprisingly, 5 at the centre also seems to draw.)

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Using that result, MiniMax our way back to the first move

So unless we can find a more aggressive opening for the second player, the whole game is a draw. Let's list all the single-move candidates for 4:

Here are some examples of the strongest lines for both players:

So, this too seems to be a forced draw. Is there any way to play 2 in order to avoid it? Playing any side just rotates the position, playing in the centre loses (see first spoiler block), playing in a far corner loses (see second and third spoiler blocks) and playing in a near corner is a draw:

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The Conclusion

So unless I made a mistake somewhere, with optimal play from both sides

(Answer to question 2: with only one move allowed per turn, the result is the same.)