1 The last exercise of Hartshorne chapter 4 section 4 proves that there is no elliptic surface over $\mathbb A^1$ with nonconstant $j$ invariant. Obviously this is a special case.

2 No as well. Put the fiber in Weirstrauss form near $j=0$. Then one can write it as $y^2=x^3-g_2x-g_3$, where $g_2$ and $g_3$ are both functions of $j$ that are well-defined at $j=0$. Furthermore clearly $g_3$ is nonvanishing at $j=0$ and $g_2$ is vanishing at $j=0$. Then computing the $j$ invariant near $j=0$ as $g_2^3/(4g_2^3-27g_3^2)$ up to a constant, we see it vanishes to at least third order, an obvious falsehood.

3 Yes, you're thinking of the Legendre family $y^2=x(x-1)(x-\lambda)$. The Legendre family is universal up to a quadratic twist - every family with full level two structure is a pullback of Legendre, up to a quadratic twist. So it is as close an approximation to the universal elliptic curve over $Y(2)$ as you can get in the category of schemes.

1 The last exercise of Hartshorne chapter 4 section 4 proves that there is no elliptic surface over $\mathbb A^1$ with nonconstant $j$ invariant. Obviously this is a special case.

2 No as well. Put the fiber in Weirstrass form near $j=0$. Then one can write it as $y^2=x^3-g_2x-g_3$, where $g_2$ and $g_3$ are both functions of $j$ that are well-defined at $j=0$. Furthermore clearly $g_3$ is nonvanishing at $j=0$ and $g_2$ is vanishing at $j=0$. Then computing the $j$ invariant near $j=0$ as $g_2^3/(4g_2^3-27g_3^2)$ up to a constant, we see it vanishes to at least third order, an obvious falsehood.

3 Yes, you're thinking of the Legendre family $y^2=x(x-1)(x-\lambda)$. The Legendre family is universal up to a quadratic twist - every family with full level two structure is a pullback of Legendre, up to a quadratic twist. So it is as close an approximation to the universal elliptic curve over $Y(2)$ as you can get in the category of schemes.