Here's one option of a type you didn't suggest.
Take any symmetric game. Modify the game as follows: After the first player's first move, give the second player the option to continue the game as usual, or to switch places, taking the first player's position and giving the (former) first player the next move as second player.
Assuming that ties are possible, best play on both sides must produce a tie. The first player will not make a first move that guarantees him a win, since if he did, the second player would just switch places and take the winning position. And nor will the first player hand the second player a win by making a bad first move. Best play for the first player is to play into a position that he can tie but not win, and then play for the tie. If the second player switches with him, he can still play for the tie.
In József Beck's book Combinatorial Games: Tic-Tac-Toe Theory, he states the following open problems ("unrestricted $5$-in a row" is Gomoku on an infinite board):
Open Problem 4.1. Is it true that unrestricted $5$-in-a-row is a first player win?
Open Problem 4.2. Is it true that unrestricted $n$-in-a-row is a draw for every $n\ge6$?
There are some relevant counterexamples in the more general setting of "positional games". The following information is from pp. 78-84 of József Beck's book.
A positional game is defined by a hypergraph $(V,E)$ where $V$ is the vertex set (or board) and $E$ is the set of (hyper-)edges (or winning sets), a collection of nonempty finite subsets of $E$. Two players take turns picking (previously unpicked) vertices; the game is won by the first player to succeed in picking all the vertices of a winning set.
The following example, attributed to Fred Galvin, illustrates what Beck calls "the Induced Extra Set Paradox". The board is $V=\{1,2,3,4,5,6,7\}$; the winning sets are $\{1,2,3\},\{1,3,4\},\{1,4,5\},\{1,2,5\},\{4,5,6\},\{4,5,7\},\{6,7\}$. This game is a draw, but it becomes a first-player win if the board is restricted to the subset $\{1,2,3,4\}$ with winning sets $\{1,2,3\},\{1,3,4\},\{1,4,5\},\{1,2,5\}$.
In Galvin's example the hypergraph is non-uniform, meaning that the winning sets are not all the same size. Here is a slightly larger example of the same phenomenon, attributed to Sujith Vijay, where the winning sets are all triples: the board is $V=\{1,2,3,4,5,6,7,8,9\}$, the winning sets are $\{1,2,3\},\{1,2,4\},\{1,2,5\},\{1,3,4\},\{1,5,6\},\{3,5,7\},\{2,4,8\},\{2,6,9\}$. This game is a draw, but the restriction to the board $\{1,2,3,4,5,6,7\}$ is a first-player win.
Beck states the following open problems concerning generalized Tic-Tac-Toe, analogous to your question 3:
Open Problem 5.2 Is it true that, if the $n^d$ Tic-Tac-Toe is a first player win, then the $n^D$ game, where $D\gt d$, is also a win?
Open Problem 5.3. Is it true that, if the $n^d$ game is a draw, then the $(n+1)^d$ game is also a draw?
Best Answer
There is discussion of this topic in volume 3 of Winning Ways (Berlekamp et al. 2003, pp 740–741). They discuss the general case of $n$-in-a-row on infinite boards. A strategy-stealing argument shows that each game is either a draw or a first-player win. It should be clear that if $n$-in-a-row is a first-player win, then so too is $m$-in-a-row for $m<n$, so the only real question is what is the largest $m$ that is a first-player win. They provide a proof that 9-in-a-row is a draw, and say “T.G.L. Zetters… recently showed that the second player can even draw 8-in-a-row”.