Has 管理 something to do with plumbing? If so then
It's hard to answer all you questions once, just one of them

1. Does a native speaker feels this connotation in 管理?


No, at least I don't.

For 管理, it has nothing to do with 管子/管道 (tube;pipe).

From 新汉英大辞典,


  管理 [guǎn lǐ]
  1. manage; run; administer; supervise; rule; administration; management; regulation:


2. Managing here is taken as a metaphore for guiding the "flow" 
in the desired direction.


Of course, broadly speaking, you could say that but literally it doesn't.

Here 's a perspective about the meaning evolution from 管 to 管理.


  “管”的实义动词义，应是由其名词义发展而来的。其语义演变过程大体是：最初指一种乐器，而后代指这种形状的东西，尤指“键”。再有此发展到管钥匙，主管库房的人，再衍生出动词义“掌管、管理”；


[ “管” 's notional verb meaning should be developed from its noun meaning. The meaning evolution is roughly: first indicates a kind of instrument, and then indicates things with this kind of shape, especially key. And then developed to person who is in charge of the key or storehouse, then derived the verb meaning of be in charge of; administer, manage. ]

So 管理 's literal sense should be administer;control;in chare of.