How did 'ply' evolve into these 4 different definitions?

Each of the four definitions "radiates" out from the original idea of plek- "to plait, twist".

If you have ever observed a person spinning thread, weaving cloth, or making a woven basket, you know that it can be meticulous, slow, and repetitious work:

• The individual elements need to be aligned properly.
• Each element must be interconnected with the other elements.
• A pattern must be carefully maintained for strength and beauty.

All of the definitions emanate from this notion of slow repetitious work:

1. Definition one of ply suggests this slow repetitious work with simple tools, as on page 101 of Alfred J. Church's Stories of the Old World

Yet it is not for Troy, or for the people, or even for my father or my mother that I care so much, as for thee in the day when some Greek shall carry thee away captive, and thou shalt ply the loom or carry the pitcher in the land of Greece.

2. Definition two of ply suggests the slow repetitious work of traveling one small step at a time before modern transportation machines made travel fast and easy. Consider page 20 of Neil Wigglesworth's The Social History of English Rowing:

London watermen brought their wherries from the capital on wagons 'to ply upon the River Cam and row people up and down from the fair to town.'

3. Definition three of ply suggests the slow repetitious work of influencing a person with one serving after another of food and drink. From page 312 in Volume 8 of Arabian Nights:

Nur al'Din refused, but the Frank ceased not to ply him with meat and drink and lure him with lucre, still adding to his offers, till he bid him ten thousand dinars for her; whereupon Nur al'Din, in his drunkenness, said before the merchants, "I sell her to thee..."

4. Definition 4 of ply suggests the slow repetitious work of interrogating someone with one question after another. From The Vernon Courier, THURSDAY JULY 4, 1889 Vol. IV, No. 5

A HINT TO MOTHERS –
If you wish to cultivate a gossiping, meddling, censorious sprit in your children, be sure when they come home from church, a visit, or any place to which you do not accompany them, to ply them with question concerning what everybody were [sic], how everybody looked, and what everybody said and did...

5. The noun definitions of ply all connect directly back to the notion of weaving and folding:

noun (plural plies)

1.0 A thickness or layer of a folded or laminated material:

1.1 [USUALLY IN COMBINATION] A strand of yarn or rope:
[AS MODIFIER]: four-ply yarn

1.2 The number of multiple layers or strands of which something is made:

2.0 short for plywood.

3.0 [MASS NOUN] (In game theory) the number of levels at which branching occurs in a tree of possible outcomes, typically corresponding to the number of moves ahead (in chess strictly half-moves ahead) considered by a computer program.

3.1 [COUNT NOUN] A half-move (i.e. one player’s move) in computer chess.

ODO