For question 1: Write $g_0$ for your base unit volume flat metric on $\mathbb{T}^2$. Fix a point $p\in\mathbb{T}^2$. Let $u_i$ be a sequence of functions on $\mathbb{T}^2$ such that :
- $\int_{\mathbb{T}^2}u_i^2dv_{g_0}=1$
- $u_i$ is constant equal to $1/i$ outside a ball (for $g_0$) of radius $\tfrac{1}{i}\to 0$ around $p$
- $u_i(p)$ goes to $\infty$ as $i\to\infty$.
- $u_i$ depends only on $d_{g_0}(p,\cdot)$
Now $g_i=u_i^2g_0$ has a long "finger" at $p_0$ attached to a tiny flat torus. And its diameter is going to $+\infty$ (it takes about $u_i(p)^2$ to go from the tip of the finger to the flat part).
For question 2 : This was wrong, I did not understand the question properly.
For question 3 : modulus is invariant under scaling so any holomorphic disk contains annuli of arbitrary finite modulus.