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}$can be arbitrarily large. 1 For (3) you can attain any ordinal$\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.