Learn English – How to avoid ambiguity when referring to a 24-hour period

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.

Quarter-hour would be your best bet, although native speakers tend not to use this in regular conversation. Quarter of an hour is also correct, but, I daresay, less likely to be used. The phrase, every fifteen minutes can often be heard in everyday usage.

For the specific software situation you mentioned, again, quarter-hour would be your best bet. On its own, quarter in temporal contexts usually refers to quarter of a year.