Unless you are a cleric or religious bound to pray the Hours, you can pray out of devotion whenever you want.
We can pray it as devotion, or we can pray it as liturgy and thereby choose to pray the Liturgy of the Hours in communion with the entire Church. In the latter case one is required to observe the verity of each hour but some flexibility is allowed.
To answer the OP’s questions:
The OOR was meant to be prayed at any time; however it’s historical root is in the office of Vigils (or Matins) of the pre-Vatican II tradition. Vigils is still part of the monastic tradition and many communities that use the LOTH instead of a monastic breviary will say it as Vigils, which is my own practice. There are even optional OT canticles in the back of the LOTH for use on Sundays, feasts and solemnities which in monastic and pre-Vatican II tradition had a “third nocturne”.
Traditionally in monastic usage, Vigils was prayed during the night, and was ended by Lauds. Originally, Lauds was simply the last three “Laudate” psalms of Vigils (ps. 148, 149 and 150), which also happen to be the last psalms of the psalter. Eventually this morphed into two distinct Offices. Keep in mind that monastic time, before the invention of the clock, was very elastic, and the Office was different in summer and winter to account for the shorter nights of summer.
In modern usage, governed by a clock, the following approximate time ranges can be used:
OOR: any convenient time (one can join it to Lauds and form a more traditional office of “Matins” where the OOR is the “vigils” part and Morning Prayer is the “Lauds” part). If using it as a traditional night Office, it can be said the preceding evening either combined with or separately from Compline, during the middle of the night, or early in the morning. Let’s assume it’s said very early in the morning. A person’s schedule could look like this:
Office of Readings: between 5 and 6 am
Lauds: between 6 and 8:30 am
Prime (first hour): abolished (but when in force, between 6 and 7 am)
Terce (third hour): between 8 and 10 am
Sext (sixth hour): between 11:30 am and 1 pm
None (ninth hour): between 2 and 4 pm
Vespers (at the lighting of the lamps), between 4 and 7 pm (it is interesting to note that the Rule of St. Benedict called for the entire office of Vespers to be said in daylight, so it would be earlier in winter, later in summer).
Compline: before retiring for the night (in monastic usage, can be as early as 6:30 pm or 9 pm depending on when the community rises for Vigils).
Mid-day prayer can be said at any one of the three canonical daytime hours of Terce, Sext or None. Moreover, if one wants to observe all of the canonical daytime hours, one can use the complementary psalmody in each volume of the LOTH (the Gradual psalms). In that case one says the mid-day hour at one of the three, and the appropriate complementary psalms at the other two.
For the structure of the Office, the hymns are to be sung, if possible, or read/recited, if not. The antiphons are said at the beginning of each psalm, and optionally, at the end as well. Moreover, if a psalm is divided into segments, and an antiphon is suggested for each segment, it is permissible to recite the first antiphon only and pray the psalm through entirely under one antiphon and glory be.
To combine the OOR to Lauds to make a more traditional office of “Matins” (Vigils+Lauds), one would proceed as follows:
Hymn (either from the OOR or Morning Prayer)
Psalmody of the OOR
Bible reading and Responsory of the OOR
Patristic or hagiographic reading and responsory from the OOR
Optionally OT canticles for Sundays, feasts and solemnities only
Te Deum (if Sunday-except Lent, feast or solemnity)
1st psalm of Lauds (Morning Prayer) then the rest of Morning Prayer.
Hope this helps. The LOTH is a beautiful and very ancient tradition that goes back to the Desert Fathers, and which even had its roots in Jewish liturgy where psalms were prayed at morning and evening. I’m always thrilled when others decide to take up this tradition in their own prayer life, and am very grateful for the current LOTH which had, as one of its intended benefits,the bringing this rich tradition into the lives of the laity, instead of being limited to the clergy and religious as prior to Vatican II.
