Do prices on airline tickets generally go down following the Christmas
and New Year holidays?

yes

Generally yes, but it depends on the destination

In the UK, tickets to skiing resorts can go up

The general rule is that you get the cheapest deal if you travel when
nobody else wants to travel - price flexing is an attempt to get
constant demand.

All bets are off when it comes to late booking

Yes, at least in the United States because there are no more big
national holidays or school vaction periods untill spring break. This
makes airlines cut their prices to make small trips to places more
inticing.