and given AJL's note in comments, I guess that 24 means
the 24th letter of the alphabet -- X, since he is King Caspian X.
Although it makes my brain hurt to interpret a Roman numeral that way.
Novice shipwright, seafarer, navigator.
The plot of Voyage of the Dawn Treader
I live in a land removed,
Yet my ancestors came from yours.
Caspian lives in Narnia which is populated with humans. More specifically as rand al'thor said in comments, "Caspian is a Telmarine, descended from pirates who slipped from this world into the world of Narnia."
My wife's name is a flower and sick,
In the books she has no name, but in the movie it is Lilliandill (lily and ill)
My tutor was straight out of old times,
His tutor Doctor Cornelius was part dwarf and thus from an older race than the humans. Also he told Caspian many things about the old times
And my son swapped my protein structure prediction with a small stream(that didn't have doubled).
This is just saying that the name Rilian is like Caspian with "casp" (http://predictioncenter.org/casproll/index.cgi - CASP ROLL - obviously the author just googled CASP and took something) replaced with "ril" ("rill" = a small stream, without the double "l")
I was the first one who they saw, an old friend.
In The Silver Chair, Jill and Eustace are supposed to greet the first person they see, who is an old friend. Eustace sees Caspian first...
But left because of the great time rift.
... but Caspian sails away unrecognized because Eustace doesn't recognize him since it's been many years in Narnia (though not in England).
Welcome to the painting, friends and company
In Voyage of the Dawn Treader the kids get to Narnia through a painting.