This is an apparently simple question which is surprisingly difficult to answer, simply because White was rather sketchy in detailing how Merlyn's memory worked. Nonetheless I do think there is enough information in the text to answer at least two of the three sub-questions at the end:
- Does Merlyn age in reverse? Does he physically grow younger as time advances like Benjamin Button?
Yes, as time passes Merlyn gets physically younger. This is addressed in Chapter XX of The Once and Future King, describing the final years of the Wart's education:
The Wart continued to be stupid, fond of Kay, and interested in birds.
Merlyn looked younger every year — which was only natural, because he
So every year that passes, Merlyn becomes a year younger. We can extrapolate this forward to his eventual birth in the Wart's far future.
- How does Merlyn experience memory? Does he see both the past, present, and future like an oracle or prophet? Or does he only see the
present and future and not the past? The above passage makes me think
that Merlyn "remembers" the future but does not remember the past.
This is probably the most confusing part of White's scheme. As the OP notes, we are given some explanation when the Wart first meets Merlyn:
"Now ordinary people are born forwards in Time, if you understand what
I mean, and nearly everything in the world goes forward too. This
makes it quite easy for the ordinary people to live... But I
unfortunately was born at the wrong end of time, and I have to live
backwards from in front, while surrounded by a lot of people living
forwards from behind. Some people call it having second sight."
Merlyn remembers Mafeking Night and he Boer War from the early 1900s, and from a passage in The Book of Merlyn he remembers ration cards ("In one of the miserable wars when I was a younger man, it was found necessary to issue to the people of England a set of printed cards which entitled them to food"), so for the sake of argument we can put his birth at 2000 AD. If we put the Wart's birth (also somewhat arbitrarily) at 1000 AD, we can graph their ages in this way:
i.e. Merlyn's age decreases with time, while the Wart's rises. The shaded areas represent the time that the two characters can in principle remember at the moment of their first meeting. That is, Merlyn can remember all future events up to his birth, while the Wart can remember past events back to his birth (assuming they have perfect recall - which Merlyn definitely doesn't).
If we know look at a chronologically later time, we see that:
So while the Wart remembers their first meeting, as is completely normal, Merlyn does not. This is described (albeit somewhat confusingly) at the conclusion of the previous passage from TOAFK:
He [Merlyn] stopped talking and looked at the Wart in an anxious way.
"Have I told you this before?"
"No, we only met about half an hour ago."
"So little time to pass?" said Merlyn, and a big tear ran down to the
end of his nose.
So indeed Merlyn has lost the memory of first meeting the Wart half an hour ago, while retaining his knowledge of events that are further in the Wart's future.
- How does Merlyn experience time? Does he live backwards literally (i.e. if I spoke a sentence to him, he would hear it backwards)? Or
does he go to sleep every night and wake up on the previous day with
knowledge of all following days (like a twist on Groundhog Day)?
This is more difficult to answer, as I do not believe that Merlyn remarks much on his subjective experience of living backwards, except to say that it was confusing (especially for verb tenses). It does appear that his memory is lost more-or-less continuously, rather than in discrete chunks as in Groundhog Day. I say "more-or-less" because, as Hypnosifi observed in comments, Merlyn does seem to be able to retain short-term memory for a while. For example, he does not forget how to finish a sentence he has started, and is able to participate in normal conversations. This seems a very short term effect though, limited to at most to thirty minutes or so (the interval in which he forgot the circumstances of meeting the Wart).
It makes for a rather depressing universe in which there is a complete absence of freewill, and all events are preordained.