I have three questions.

1. I consider a sequence of metrics $h_n$ on a two-dimensional torus which all induce the same conformal structure. Suppose that the volume of $h_n$ is always $1$. Is it possible that the diameter of $h_n$ tends to infinity?

2. Consider such a sequence of metrics along which the diameter is bounded but whose conformal class is allowed to vary. Is it possible that the associated sequence of conformal classes diverges? (the moduli space of conformal classes on the torus is $\mathbb{H}^2/ \mathrm{SL}_2(\mathbb{Z}))$ and by diverging I mean escaping from every compact set of this moduli space).

3. Fix a conformal structure on a torus. Can I holomorphically embed cylinders of arbitrarily large modulus in this torus in a $\pi_1$-injective way?

Thanks for your attention.

Selim