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Is there a naming convention for temporary moons? Or do they follow the naming conventions of other small bodies?
In the case of 2006 RH$_{120}$, 2015 HP$_{116}$, 2020 CD$_3$ and 2020 SO, the first number is the year. I assume the following letter+number combination somehow indicates the time of year of discovery and/or the number of similar bodies discovered prior?
How were YX205B9, WT1190F, S509356 and XC83E0D named?
These are just the standard minor planet designations.
As described on the MPEC page on New- and Old-Style Minor Planet Designations, the new-style provisional designations for minor planets (like 2006 RH120) are in the form year, letter indicating the half-month, letter indicating the sequence within the half month, and number indicating the cycle of the second letter (no number = 1st cycle, 1 = 2nd cycle, etc.)
The letters for half-months omits the letter I and doesn't use Z as there are only 24 half-months in the year. The second half-month in a month starts on the 16th for all months.
The letter indicating the order is the alphabetic sequence, again skipping I but this time including Z.
Old-style provisional designations were used in the 19th century and are not relevant here.
These are assigned by observatories according to their own rules, rather than by the MPEC. I think three of the ones you list (YX205B9, WT1190F and XC83E0D) were assigned by the Catalina Sky Survey, but I haven't seen any documentation of what their internal rules for assigning identifiers is. These objects were confirmed to be artificial before they had sufficient orbit determinations to be assigned minor planet designations, so never received provisional designations.
Provisional designations such as 2020 CD3 are as you inferred: the number and first letter indicate the year and half-month of discovery, and the remainder is sequential within that half-month. Well-observed objects with well-determined orbits get a permanent number designation and become eligible for a name, e.g. (101995) Bennu was previously known as 1999 RQ36.
Temporary designations of unknown objects are left up to the reporting observer, within certain guidelines: use 6 (preferred) or 7 characters beginning with a letter, avoid prefixes like J99 and K18 which resemble packed provisional designations, and try to make it unique. Objects listed on the NEO confirmation page use such identifiers until there are enough observations for an orbit and a provisional designation.
If an object is known by a temporary designation long after discovery, either there weren't enough observations to determine a useful heliocentric orbit, or it was identified as space junk. Occasionally a provisional designation is retracted, as when 2015 HP116 was identified as the Gaia spacecraft.
