What exactly is a liber ordinarius?

The Catholic Encyclopedia reports:

In the Middle Ages, and indeed almost down to the invention of printing, the books used in the service of the Church were much more divided up than they are at present. Instead of one book, our modern Breviary for example, containing the whole Office, we find at least four books — the Psalterium, the Hymnarium, the Antiphonarium, and the Legendarium, or book of lessons, all in separate volumes. Rubrics or ritual directions were rarely written down in connexion with the text to which they belonged...

The Encyclopedia starts by explaining the situation regarding Liturgical books (that is to say, for Holy Mass and the Liturgy of the Hours) through the Low Middle Ages. Since both parchment and manpower was expensive, liturgical books usually ommitted the rubricated directions for priests and only recorded the words for reciting, and were also split into smaller volumes, easier to copy.

However, about the eleventh century there grew up a tendency towards greater elaboration and precision in rubrical directions for the services, and at the same time we notice the beginning of a more or less strongly marked division of these directions into two classes, which in the case of the Sarum Use are conveniently distinguished as the Customary and the Ordinal.

Eventually, as Europe recovered from the Dark Ages, as well as the Carolingian imperative towards more uniformity in worship, these rubrics were collected and spelled out in Liturgical books which simplified the preparation work for the priest, as well as minimised the possibility that a less well trained priest might mix up directions and celebrate wrongly. These compendia of Liturgical directions were collected into two classes, one more general ("Customary" in the case of the Sarum Use) and one more specific ("Ordinal" in the same Use).

It is out of the latter of these books, i.e. the Ordinal (often called Ordinarium and Liber Ordinarius), that the "Directorium", or "Pye", and eventually also our own modern "Ordo recitandi" were in due time evolved.

Here we see the connection between your term Liber Ordinarius and the Sarum Ordinal (which was not limited to Sarum, and was produced by other dioceses), which eventually evolved into "our own modern" (that is, pre-Vatican II, because by "modern" we mean "1913") Ordo recitandi.

As for how the Ordo recitandi is laid out:

It consists simply of a calendar for the year, in which there are printed against each day concise directions concerning the Office and Mass to be said on that day. The calendar is usually provided with some indication of fast days, special indulgences, days of devotion, and other items of information which it may be convenient for the clergy to be reminded of as they occur.

The Liber Ordinarius seems to have taken much the same shape, a calendar where the priest looks up all the special rubrics which must be followed in a given day.

Note that nowadays the Mass of Paul VI seems to take a different approach, letting the priests queue up the correct readings for the day by using a system of several bookmarks out of the same single book, either the Ordo Missae for Holy Mass or the Breviarium for the Liturgy of the Hours, which are prepared beforehand before starting Liturgy.

In conclusion, regarding your points:

1. Does it cover the entire liturgical year?

   Yes, libri ordinarii were conceived to be a handy reference to the rubrics which were to be used in a given diocese or group of dioceses for a single year.

2. Do I remember correctly that it may contain only incipits of the to be used texts/chants?

   Yes, they had (as the Encyclopedia puts it) "concise directions" regarding Office and Mass, which means the text was generally abbreviated, though enough of the text was given that a priest would have no doubt as to which specific forms were to be used, whenever there was a choice of two or more.

3. If so, are there ones that contain the full texts/chants?

   Not libri ordinarii, although there were other books, like the Ordo missae, legendaria, which together were able to specify the entirety of the Liturgy in those times. It's important to remember that this is pre-printing press, so these books were extremely expensive to produce; therefore a single, "all in one" version of the books was prohibitively expensive. This eventually changed from the 15th Century on, though.

4. Are they still in use today or did they get replaced by something else?

   They eventually evolved into the Ordines recitandi which were used until the general reforms in 1970 after Vatican II, and were basically the same thing — a book with an entry for each day of the year and which specific readings, antiphons, psalms, and proper prayers were due for that day.

   However, when the most recent Liturgical reform was promulgated, the rubrics were collected into single books, the Ordo missae for Holy Mass and the Breviarium for the Liturgy of the Hours. Note, however, that a Traditionalist group under a company called Saint Lawrence Press produces Ordines Recitandi for the current year, according to the pre-Vatican II Missal.