The first thought I had in reflecting on how to approach these questions led me to the first two lines of verse 6.80 of the Madhyamakāvatāra, which appears on p. 279 of The Karmapa's Middle Way:
The conventional truth is the method;
The ultimate truth is what arises from the method.
In terms of the personal experience of traversing the path, I think this couplet summarizes the utilitarian aspect of the two truths very nicely. The conventional (or relative) truth is composed of everything we engage in in order to develop further insight into the nature of things. The ultimate truth is the wisdom that arises from those engagements.
Ontologically, the meaning of the two truths is summarized in one of Chandrakīrti's most famous verses, verse 6.23, which is found on page 204:
Since all things can be seen genuinely or falsely,
Every thing bears two natures.
The Buddha taught that the object of genuine seeing is suchness
And that false seeing is the relative truth.
It should be noted that Ari Goldfield made a beautiful song arrangement for this verse, with which many of you will be familiar. The translation for that arrangement reads:
There are two ways of seeing everything,
The perfect way and the false way.
So each and every thing that can ever be found
Holds two natures within.
And what does perfect seeing see?
It sees the suchness of all things.
And false seeing sees the relative truth.
This is what the perfect Buddha said.
I think as a general rule of thumb in reading The Karmapa's Middle Way, it is helpful to be mindful of the distinction between Wangchuk Dorje's explanation of the basic principles of topics versus his unique presentations of those topics. For his basic presentation of the two truths, reading the first five paragraphs of page 205 in the book should be helpful. In sum, Wangchuk Dorje seems to say that relative truth includes everything that is perceived and experienced by a mind that has not realized the true nature of things, and, indeed, by a mind that is not currently abiding in the realization of the true nature of things (though in a future posting I will detail the distinction between relative truth and the mere relative, a distinction within which the "mere relative" would form a unique qualification to this definition). The ultimate truth is what is seen by noble beings, i.e. by those who are currently abiding in a realization of the way things really are. In other words, relative truth is what is seen by a mistaken mind; ultimate truth is what is seen by an unmistaken mind.
Given these definitions, it is therefore straightfoward to determine how to engage the relative truth in day-to-day life, because the relative truth is all that our minds can reach at present (please refer to my response to Nick's comments for a fuller explanation of the "realities" that are within the grasp of ordinary beings in a context of sūtrayāna): from the strict perspective of Chanrakīti's and the Karmapa's ontology, whatever we are doing on the path, that is relative truth.
Can we engage in the ultimate truth at all as ordinary beings? No and yes. We have already looked at why no is the answer, but we can study and talk about the ultimate truth from a conceptual vantage point. The Nyingma school makes frequent use of a way of categorizing the ultimate that is very helpful in this context: the ultimate, they say, is of two types: 1) the nominal ultimate truth, and 2) the genuine ultimate truth. The genuine ultimate truth is exactly "what perfect seeing sees." For everyone else, there is the nominal ultimate truth. The nominal ultimate truth is not the real deal, but it is helpful in that it helps us to form conceptual images of what the ultimate truth is like. These conceptual images are very important, because they form the basis of certainty that can become further refined until certainty itself serves as a springboard for direct realization. In his discussion of the paths and levels in the Treasury of Knowledge, Jamgön Kongtrul Lodrö Thaye talks about the extremeley clear "object generality" of emptiness that the meditator forms on the path of juncture, the second of the five paths. This object generality is not genuine emptiness, but it is the fruit of intensive training in the concept of emptiness, and it is the final, most refined form of conceptual emptiness that is present before the final veils to direct realization are broken through. So gaining conceptual understanding of ultimate truth is very important: all our studies and contemplations of the ultimate are, strictly speaking, relative truth, but without them, there would be no way to approach the actual ultimate.
In this vein, the Karmapa, on pages 48 and 279 of the book, quotes the ever-important verse from Nāgārjuna:
Without relying on conventions,
One cannot realize the ultimate.
Without realizing the ultimate,
One cannot attain nirvāṇa.
In the final analysis, anything that depends on another concept is a relative truth. Therefore, ultimate truth in the context of the pair of relative and ultimate is nothing more than a relative truth. To paraphrase a quote from the Buddha, in true reality, there is only one truth, and that truth is the ultimate truth. Yet, of course, from the point of view of the Karmapa's thorough analysis, even that rendition of the ultimate truth would need to be released by the meditator aiming to dwell in the ultimate equipoise.
This is a very short, and undoubtedly severely truncated, description of the way the two truths play out in the Karma Kagyü Middle Way tradition (or at least the Middle Way tradition of the Eighth and Ninth Karmapas).
I want to stress again, for the general reader, that I think it is much more important in the beginning to learn and become familiar with the basic concepts of the two truths before one worries too much about the subtleties of Wangchuk Dorje's unique presentations.
A quick final thought on the two truths in meditative practice: roughly speaking, one can consider the first five pāramitās to be practices of the relative truth and the sixth pāramitā to be a practice of the ultimate truth. In our analytical meditation and in our lives, we employ as subjects of analysis the phenomena--our phenomena--of relative truth and reflect on their deeper nature using the teachings' scripture and reasoning.