Primarily, the impurities are of two types:

1. Nijaasat e Haqeeqiyyah physical impurities.

2. Nijaasat e Hukmiyyah state of being impure, when there is a need of Wudhu or Ghusl.

Nijaasat e Haqeeqiyyah is also of two types:

Galeezah (Heavy filth and impurity)

Khafeefah (Light filth and impurity)

Furthermore, if the former (heavy impurity and filth) remains on a person’s body, clothes, etc. while he/she is praying, its amount is taken into consideration. Any solid filth weighing more than 4.86 grams, and any liquid more than the amount that spreads over a person’s inner circle of the palm (2.75cm diameter), invalidates the prayer.

As far as the latter is concerned, if more than one-fourth of a limb or one-fourth of one's clothes are soiled with Khafeefah (light impurity and filth), it will render the Salah invalid. Although, less than the aforementioned amounts are overlooked by Shariah; yet, it is disliked to pray with it. (Tataawa Uthmani Vol. 1 Pg. 282) 

Additionally, there are two types of Najaasat e Haqeeqiyyah:

Marieyyah (Visible filth)

Ghair Marieyyah (Invisible filth)

The ruling with regards to cleaning and purifying these is: in Marieyyah (visible impurity), one is required to remove the actual substance, unless such traces remain which are difficult to remove. And, in Ghair Marieyyah (invisible filth), one is required to wash the impure part of body thrice and, in clothes, one is also required to wash thrice and squeeze the impure part of the cloth after each wash. (Nurul Idaah Pg. 59-62) 

In the light of the above, to clean and purify one’s body from Madhi (a white sticky prostatic fluid) he/she has to wash the body part thrice since it is an invisible heavy filth; beside this, there is no need to dry the washed part of the body after every washing. (Fataawa Hindiyyah Vol. 1 Pg. 42) 

It is not necessary to use soap for cleaning and purifying one’s body, clothes, etc.; rather, it is only advised and meritorious to use soap. (Majma'ul Anhur Vol. 1 Pg. 60) 

Lastly, as enquired in the query, if the water drips on the other parts of the body whilst cleaning the impurity, the other portions will become impure. However, if one washes the entire impure limb thrice in such a manner that the water does not splash on the other limbs or the water flows off, the other limbs will not become impure. (Fataawa Mahmudiyyah Vol. 8 Pg. 269)