Why is the character 漢 built in such a weird way? 漢 is the character meaning "China" or "Sino-". 

It supposedly ultimately derives from the name of the river Han. I understand why the water radical is present there, but I have no idea why the seemingly random characters 中, 人, 廿 and some two horizontal lines between 中 and 人 are combined and why were these particular ones chosen to represent the river's name.
According to 说文解字,


  从水，難省聲


So 漢 is taking  as its pictophonetic part.

1. What does  mean?

This is the ancient inscriptions (甲骨文 and 金文) of it (from 字源谈趣).



Its original meaning is to fire someone (as a penalty), who is tied up and crying to the sky. So it's used for the meaning of suffering or disaster, and then 難, 艱, 嘆 were created based on it.

2. Why 漢 took this character as its pictophonetic part?

1st perspective, from 字源谈趣,


  漢水的“漢”，以“氵”（水）表意，以“漢（去氵）”表音兼表义。意指古代聚居在漢水一带中原地区的我国最大的民族——华夏族（后发展以“漢”为名的漢族），是历经千灾万难，在与艰难困苦斗争中发展起来的。


[“漢” is taking “氵”（water） as its ideographical part, taking “漢”（getting rid of 氵） as its pictophonetic and ideographical part. It indicates the largest nationality residing at 漢水 struggled to grew up (into the han nationality).]

2nd perspective, from 象形字典,


  造字本义：长江最大支流，古代中原朝庭常常流放政治犯的西北水域。


[In ancient times 漢水 served as a place of banishment for criminals.]