RE
There are a few minor inaccuracies in your discussion on the calendar
“whereas Jewish holdays are determined through the moon. Pesach is spring and Succos fall time”
This is a bit contradictory if holidays are by the moon then they have nothing to do with seasons. The truth is holidays are based on BOTH the moon and the sun. The DATE is based on the moon but has to be in right season based on the sun.
“that lunar year having normally 354 days”
There isnt really such a thing as a “lunar year*” we take 12 months and call it a year these years can vary 353, 354, 355 years of course there are leap years with 13 months with 383, 384, 385 days. I’m not sure what you mean by “normally” having 354 days. This isnt the most common length for a year (mode) nor the average (mean) . The average length for a year (rounded to the nearest whole number) is actually 365 exactly as you said to keep it in sync with the sun.
This is why the Gemara in Makkos says Hasatan = 364 corresponding to the days of the year minus Yom kipppur. Many wonder why use the “goyish calendar” or the “solar calendar” The answer is simple. The number of days in an (average) Jewish year is 365 (rounded to the whole number). There is no such thing a “Days in a lunar year” A lunar year doesn’t have days it has 12 lunar months . For example see Megillah 5a Where we learn “lechodshei hashana” that we count months for a year not days. see RAshi there. If we DID count days the number of days in a year is 365 and if someone made a neder not to drink wine for a year if you count by days would have to wait 11 days longer . But we DON’t count days we count Months so after 12 months can drink wine again. it might be 3 53, 354 or 355 days but we dont count days.
“The variation of the two months above causes the movement of Rosh Hashanah as Adu lo Rosh,”
The opposite is true Lo Adu Rosh (and other dechuyos) determine how many days are in kislev and cheshvan . the Molad for Tishrei 5782 is Monday Night. Because the molad is after noon Rosh Hashana is pushed to Tuesday. Because Rosh hashana is pushed to Tuesday (not pushed because Lo adu rosh, but principle is the same), and thsi past Rosh Hashana was Shabbos we need a calendar with 353 days So we make both cheshvan and Kislev chaser with 29 days “subtracting” a day from the 354 days we get by having 6 months of 29 alternatign with 6 months 30 ( (6×30)+(6×29) = 354).
The reason why we don;t want Rosh hashana on Friday or Wednesday is to avoid Y”K on Sunday or Friday which see R”H 20a
(* Similarly, there isnt really such a thing a solar month. We take a solar year (approximately 365 days) and divide it into 12 arbitrary chunks of time varying 31/30 days (and one 28) )
These are admittedly minor “inaccuracies” and forgive my nitpicking. I just enjoy the topic