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For (3) you can attain any ordinal $\alpha < \omega_2$. This can be shown by using transfinite induction. As an induction hypothesis on $\alpha$ assume that for any strictly increasing $f:\mathbb{N} \to \mathbb{N}$ there is a sequence $f_\xi$ for ${\xi\in\alpha + 1}$ ordered by eventual domination such that $f_\alpha = f$. The first case to consider is $\alpha = \omega_1 + 1$. Construct $f_\xi$ for $\xi\in\omega_1$ by using a countable induction at each stage to fill in the gap between $f_\xi$ and $f= f_{\omega_1}$.

Now given an arbitrary $\alpha < \omega_2$ either $\alpha = \beta+1$ --- in which case take $f'= f/2$ and add $f$ to the well ordered chain of length $\beta$ ending with $f'$--- or $\alpha$ is a limit. If $\alpha$ is a limit of cofinality $\omega_1$ and $f$ is given start with a chain $f_\xi$ ordered by eventual domination of length $\omega_1$ \omega_1+1$ ending with $f$ and choose a cofinal sequence $\alpha_\xi$ for ${\xi\in\omega_1}$ cofinal in $\alpha$. Then use the induction hypothesis to fill in the interval between $f_{\alpha_\xi}$ and $f_{\alpha_{\xi+1}}$ with a chain of order type $\mu $ such that $\alpha_\xi + \mu = \alpha_{\xi + 1}$. The countable cofinality case is similar.

This is the best that can be done because in the model obtained by adding $\aleph_3$ Cohen reals to a model of CH there are no chains of length $\omega_2$. Of course there are always chains of length $\mathfrak{b}$ and $\mathfrak{b}$ can be arbitrarily large.
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For (3) you can attain any ordinal $\alpha < \omega_2$. This can be shown by using transfinite induction. As an induction hypothesis on $\alpha$ assume that for any strictly increasing $f:\mathbb{N} \to \mathbb{N}$ there is a sequence $f_\xi$ for ${\xi\in\alpha + 1}$ ordered by eventual domination such that $f_\alpha = f$. The first case to consider is $\alpha = \omega_1 + 1$. Construct $f_\xi$ for $\xi\in\omega_1$ by using a countable induction at each stage to fill in the gap between $f_\xi$ and $f= f_{\omega_1}$.

Now given an arbitrary $\alpha < \omega_2$ either $\alpha = \beta+1$ --- in which case take $f'= f/2$ and add $f$ to the well ordered chain of length $\beta$ ending with $f'$--- or $\alpha$ is a limit. If $\alpha$ is a limit of cofinality $\omega_1$ and $f$ is given start with a chain $f_\xi$ ordered by eventual domination of length $\omega_1$ ending with $f$ and choose a cofinal sequence $\alpha_\xi$ for ${\xi\in\omega_1}$ in $\alpha$. Then use the induction hypothesis to fill in the interval between $f_{\alpha_\xi}$ and $f_{\alpha_{\xi+1}}$ with a chain of order type $\mu $ such that $\alpha_\xi + \mu = \alpha_{\xi + 1}$. The countable cofinality case is similar.

This is the best that can be done because in the model obtained by adding $\aleph_3$ Cohen reals to a model of CH there are no chains of length $\omega_2$. Of course there are always chains of length $\mathfrak{b}$ and $\mathfrak{b}$ can be arbitrarily large.