How many orders of infinity are there?
For asymptotic domination, commonly denoted ${\leq^*}$ and often called eventual domination, this has been answered by Stephen Hechler, On the existence of certain cofinal subsets of ${}^{\omega }\omega$, MR360266. What you call a complete set is usually called a dominating family.
As a poset under eventual domination, a dominating family $\mathcal{F}$ must have the following three properties:
- $\mathcal{F}$ has no maximal element.
- Every countable subset of $\mathcal{F}$ has an upper bound in $\mathcal{F}$.
- $|\mathcal{F}| \leq 2^{\aleph_0}$
Hechler showed that for any abstract poset $(P,{\leq})$ with these three properties, there is a forcing extension where all cardinals and cardinal powers are preserved, and there is a dominating family isomorphic to $(P,{\leq})$.
In particular, one can have a wellordered dominating family whose length is any cardinal $\delta$ with uncountable cofinality. In this case, the restriction $\delta \leq 2^{\aleph_0}$ is inessential since one can always add $\delta$ Cohen reals without affecting conditions (1) and (2). However, for arbitrary posets, condition (2) could be destroyed by adding reals.
The total domination order is more complex. One can always get a totally dominating family $\mathcal{F}'$ from a dominating family $\mathcal{F}$ by adding $\max(f,n) \in \mathcal{F}'$ for every $f \in \mathcal{F}$ and $n < \omega$. Since $\mathcal{F}$ is infinite, the resulting family $\mathcal{F}'$ has the same size as $\mathcal{F}$. Howerver, there does not appear to be a simple combinatorial characterization of the possibilities for the posets that arise in this way.
Francois Dorais cited the paper of Stephen Hechler that (more than) completely answers the first part of the question. For the second part, concerning other ways to view $\mathfrak d$, two other papers of Hechler are relevant; here are the MathSciNet citations:
MR0369078 (51 #5314) Hechler, Stephen H., A dozen small uncountable cardinals. TOPO 72---general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), pp. 207--218. Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974.
MR0380705 (52 #1602) Hechler, Stephen H., On a ubiquitous cardinal. Proc. Amer. Math. Soc. 52 (1975), 348--352.
Four (if I remember correctly) of the 12 cardinals in the first paper turn out to equal $\mathfrak d$, which is also the "ubiquitous cardinal" of the second paper.
Let me also mention that, if one just wants to answer the first part of the question, one doesn't need the very detailed information given by Hechler's theorem. In order to get a model of set theory with prescribed values for $\mathfrak d$ and for the cardinality $\mathfrak c$ of the continuum (subject to the necessary restrictions that both have uncountable cofinality and that $\mathfrak d\leq\mathfrak c$), it suffices to start with a model of the generalized continuum hypothesis (e.g., G\"odel's constructible universe), adjoin as many Cohen reals as the cardinal you want to be $\mathfrak d$, and then adjoin enough random reals to bring $\mathfrak c$ up to the desired value.
The forcing method introduced by Hechler in the paper that Francois cited has become one of the standard tools in the study of cardinal characteristics of the continuum. For just one example, see
MR0780528 (86i:03064) Baumgartner, James E.; Dordal, Peter, Adjoining dominating functions. J. Symbolic Logic 50 (1985), no. 1, 94--101.
Finally, let me indulge in a bit of self-promotion. On the set theory page of my web site,
http://www.math.lsa.umich.edu/~ablass/set.html ,
the first two papers are about cardinal characteristics of the continuum. The first is a short (6 pages), general-audience introduction (based on a talk at a conference for Ryll-Nardzewski's 70th birthday), and the second is a long chapter which (contrary to the "to appear" on the web site) has now appeared in the Handbook of Set Theory.
François has given an excellent answer to this question.
What you call a cofinal collection, a family $\cal F$ such that every function is dominated by a function in $\cal F$, is known as a dominating family. This is different, for example, from the similar concept of an unbounded family, a family $\cal F$ such that no function dominates every function in $\cal F$, since in partial orders as opposed to linear orders the notions of dominating and unbounded are not the same. As there are several inequivalent but similar-sounding notions here, it seems worthwhile to use the established terminology.
As Kristal mentions, I mention in this MO answer, which is also a direct answer to this question, that the dominating number d is the size of the smallest dominating family of function, the smallest family of functions such that every function is dominated by something in the family. As you point out, this number is always uncountable and at most the continuum, but as François mentioned, the particular value of d can be exactly controlled by forcing. In particular, it can achieve desired intermediate values, when CH fails.
The similar-sounding but actually inequivalent bounding number b, in contrast, is the size of the smallest unbounded family $\cal F$, a family such that no function dominates every function in $\cal F$. Since every dominating family is unbounded, it follows that b $\leq$ d. Remarkably, however, it is consistent that b $\lt$ d, and this is proved again by forcing.
There are dozens of other similar cardinal characteristics of the continuum, some of which I mention in this MO answer. For examples, researchers consider the additivity of the Lebesgue-measure, the additivity of the meager ideal, the cofinality of the symmetric group $S_\omega$ (the smallest number of proper subgroups forming a chain whose union is the whole group), the covering number (fewest number of measure zero sets to cover the reals) and variations, and so on. Researchers in this area classify and separate these different cardinals into hierarchies, and some prominent relationships are expressed by Cichon's diagram. It is often particularly desired to control some of the cardinal characteristics by forcing, while leaving others fixed, and some of the most valuable results here are general theorems that make such a conclusion. Andreas Blass, now here on MO, is one of the world experts in this area.