Missing words in English
Solution 1:
For your first point I found words such as hemeral and circadian, though there are more concept centered around a 24-hour period, but simply using the word day, as suggested by @FX_ would be the way to go, although ambiguous, but there doesn't seems to be another "English" word which would fit the bill.
On the second point, you have the word solidary, but also solidarily, which is an adverb defined as "showing solidarity". As such, we can safely assume that solidary is the adjective for the same concept.
EDIT: As for the fact that solidary against sympathetic, sympathetic comes from the Latin sympatheticus and sympathy has its own Latin words sympathia, both of each having Ancient Greek origin (sympathetikos and sympatheia), while both solidary and solidarity comes from the French solidaire which itself comes from the french word solide which has its roots in the Latin solidus, but this time not in the Greek.
Solution 2:
day (noun) 1. a period of twenty-four hours as a unit; if you want to refer to the natural rhythm of 24 hour period, the adjective is circadian
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solidary (adjective) (of a group or community) characterized by solidarity or coincidence of interests
- “Judge imposes ‘solidary bail’ for poor respondents”
- “Incentives for a solidary globalization”
- “Repression and solidary cultures of resistance”
Solution 3:
Day is the English term for the 24 hour period of a calendar. Night is the dark part, and Daytime is the light part. Daytime is then shortened to Day when the context is clear which you are talking about.
Solution 4:
Day is one of the time extent words that have both Calendric and Non-Calendric senses. The non-calendric sense refers to any 24-hour period, regardless of starting point, while the calendric sense goes like calendar numbering, from midnight to midnight. So day is the right word, provided it's used in the right constructions.
As Fillmore explains it in the Deixis Lectures:
"When nature provides sequentially recurring event types having apparently the same duration, these event types can be used to provide measuring units for temporal extent. The recurring event types that are most constant and most common and most accessible to ordinary observers are the daily alternation of light and dark, changes in how the moon looks to us, and the apparent annual course of the sun accompanied by the regularly recurring changes in the seasons.
"These particular event types are cycles which do not involve the sequencing of discrete separable events, and so, when they are used for providing units of measure, it is necessary to identify recurrences of the same phase of the cycle. Those phases which seem to have constant temporal extenst between successions of them are, for example, the full moon, the most vertical position of the sun, the shortest day of the year, etc.
"If these cycles are to be taken only as units of measure, it makes no difference which phase of the cycle is taken as the starting point for the measurement. If, however, these events are to provide concepts for locating events in "absolute time", then there is a special need for fixed-phase units, time units which have been assigned fixed starting points recognizable, in principle, by all members of the speech community. Time measure periods taken only as units of measure we can call Non-Calendric. Time measure periods having fixed starting points can be called Calendric.
"Many of the time measure words in English have both calendric and non-calendric uses, for example, the word year. If I say that the time between noon on June 28, 1972 and noon on June 28, 1973 is one year, I am using the word year non-calendrically. On the other hand, if I use the expression last year, meaning the period of time between the beginning of January 1, 1970 and the end of December 31, 1970, I am using the word year in its calendric sense."
- from Lecture 3: Time, pp 33-4