Learn English – Adjectives with Latin etymology when noun has non-Latin etymology

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.

It was believed that epilepsy seizures were triggered by moonlight hence lunatic was used for those patients.

Luna gives the adjective lunaticus. This appears in the Vulgate (405) of the Dalmatian Christian writer Saint Jerome (Eusebius Hieronymus, 348–420) as an epithet for “a moon-struck” person, whence “crazed, insane, lunatic.” It was used of epilepsy, from the notion that the seizures were precipitated by moonlight. The paroxysmal nature of the disease was thought to be dependent upon the phases of the moon.

Lexicon Orthopaedic Etymology

There is also a scientific publication titled The disease of the moon: the linguistic and pathological evolution of the English term "lunatic" from ncbi.nlm.nih.gov that the origin of the word is related to epilepsy:

The original meaning of the term “lunaticus” is not related only to insanity. In particular, its first use is documented in the Vulgate, the fifth-century Latin version of the Bible,translated from the Ancient Greek by Jerome (347–420) on commission of Pope Damasus.

In the Gospel of Matthew (17: 15–18), a father asks Jesus to cure his son because he is “lunaticus” (“Domine, misere filio meo, quia lunaticus est, et male patitur: nam saepecadit in ignem et crebro in aquam. [ . . . ] Et increpavit illum Jesus et exit ab eo daemo-nium et curatus est puer ex illa hora”). This episode is translated in the Bible of King James (1611) as follows: “Lord, have mercy on my son: for he is lunatick, and sore vexed; for ofttimes he falleth into the fire, and oft into the water [ . . . ] And Jesus rebuked the devil; and he departed out of him: and the child was cured from that very hour.” When this passage is compared with the other synoptic gospels (Luke 9: 37–43; Mark 9: 17–29), the most accurate description of the same episode leads us to understand that the boy is affected by epilepsy.

The term “lunaticus est” is the Latin translation of the Greek verb “ σεληνιαζεται ” (“seleniazetai”), which includes the prefix selen- (from σεληνη - the ancient Greek word for the moon). Therefore, the original meaning of the term “lunatic” seems to be linked to epilepsy, rather than insanity.

You can read about the legal category of lunacy and the history of the word in psychiatry in the book The Moon and Madness (By Niall McCrae).