Yes, this is possible, if you define the order to be dominance with finitely many exceptions. So f < g iff the set of n with f(n) > g(n) is finite. What you call a complete system of growth functions is called a dominating subset of $\omega^\omega$ (and a scale if it is well-ordered). See van Douwen's paper "The integers and topology" in the Handbook of Set Theoretic Topology. The minimal cardinality of such a dominating family is called $\mathfrak{d}$ in the set-theoretic literature and it's own one of the so-called cardinal invariants of the continuum. What is known is that its cofinality is at least $\mathfrak{b}$ where the latter is the minimal size of an unbounded set in $\omega^\omega$ in the partial order of eventual dominance. Also, $\mathfrak{d}$ is equal to the minimal size of a cofinal subset of $\omega^\omega$ in the total dominance order that you defined. So indeed, the problem is the same for both orders, and both have minimal size $\mathfrak{d}$. The eventual dominance is more commonly used though, and that's how I knew it at first.
This cardinal can assume almost any value (under said restriction on the cofinality at least) and there has been a lot of study on this and similar cardinal invariants and their interrelations. We can have $\omega_1 = \mathfrak{d} < \mathfrak{c}$, $\omega_1 < \mathfrak{d} < \mathfrak{c}$ and $\omega_1 < \mathfrak{d} = \mathfrak{c}$, in different models of ZFC.
Yes, this is possible, if you define the order to be dominance with finitely many exceptions. So f < g iff the set of n with f(n) > g(n) is finite. What you call a complete system of growth functions is called a dominating subset of $\omega^\omega$ (and a scale if it is well-ordered). See van Douwen's paper "The integers and topology" in the Handbook of Set Theoretic Topology. The minimal cardinality of such a dominating family is called $\mathfrak{d}$ in the set-theoretic literature and it's own of the so-called cardinal invariants of the continuum. What is known is that its cofinality is at least $\mathfrak{b}$ where the latter is the minimal size of an unbounded set in $\omega^\omega$ in the partial order of eventual dominance. Also, $\mathfrak{d}$ is equal to the minimal size of a cofinal subset of $\omega^\omega$ in the total dominance order that you defined. So indeed, the problem is the same for both orders, and both have minimal size $\mathfrak{d}$. The eventual dominance is more commonly used though, and that's how I knew it at first.
This cardinal can assume almost any value (under said restriction on the cofinality at least) and there has been a lot of study on this and similar cardinal invariants and their interrelations. We can have $\omega_1 = \mathfrak{d} < \mathfrak{c}$, $\omega_1 < \mathfrak{d} < \mathfrak{c}$ and $\omega_1 < \mathfrak{d} = \mathfrak{c}$, in different models of ZFC.