How to mention First Name, Middle Name, Last name?

In case of long names it sometimes becomes difficult to divide them in first, middle, and last parts but it can be done in most of the cases. One of the ways to do it is put one prominent name as frst name and then put the one which is on the name of family or village or tribe as the last name (generally known as surname). Rest can go under middle name.
Let us take the first example in your post that is - Vutukuru Siva Santosh Kumar.
As per my understanding we can keep Santosh as the first name, Vutukuru as the last name and the remaining one that is Siva Kumar as middle name. So I will write it as -
Santosh / Siva Kumar / Vutukuru.
In case Kumar is your surname then replace Vutukuru with Kumar.
One of the examples in your post is GVSRSR Pawan Kumar. I will like to format it as -
Pawan / GVSRSR / Kumar.
If your surname is not Kumar but is actually one of the letters (say S) in the middle name then remove it from middle name and expand it (S......) and write as -
Pawan / GVSRR Kumar / S.....
So whatever way you decide it can be done and it is better to keep one single word under first as well as under the last name. Most of the places the name data is called from surname (last name) or first name only. There should be clarity in that. Once selected we should keep it in same sequence and not change it.

The first name will be the name that is given to you. The last name is the name of your family which is known as a surname. All other names will be middle names.
For example, my name is N V Srinivasa Rao.
I take my first name as RAO, N is the Surname so it can be written in the last. V and S are the middle names. So my short name can be written as Rao, VS, N.
In the same way, you can write all the names you have mentioned above.
1. Vutukuru Siva Santosh Kumar
In the above name, Vutukuru is the surname, Kumar is the first name and Siva and Santosh will be the middle names. So you can write the above name as Kumar, Siva Santosh, Vutukuru.
Some may have no middle names and some may have more middle names.
2. Vutukuru Satyanarayana
In this name, there is no middle name. Satyanarayana is the first name and Vutukuru is the family name which is the last name. So it can be written as Satyanarayana Vutukuru.
3. Vutukuru Siva Vamsi, Here the middle name will be Siva. The last name will be Vamsi. The surname is Vutukuru. So we can write the name as Vamsi Siva Vutukuru.
4. Vutukuru Siva Priya Krishna Kumari. Here there are three middle names. They are Siva, Priya, and Krishna, the First name will be Kumari and the surname will be Vutukuru. So the name in the required order will be Kumari, Siva Priya Krishna Vutukuru.
5. G V S R S R Pavan Kumar
In this name, I have assumed that G is the family name. There are many middle names. They are V, S, R, S, R and Pavan. Now the name will be Kumar, V S R S R Pavan, G.

Please note that we can write our names the way we wanted. But keep in mind that always family name will be the last name. First and middle names can be of our choice. But always we should maintain the same sequence.
If N V Srinivasa Rao.
I can mention this as Rao, VS. N .
otherwise, I can also mention the same as
Srinivasarao, V. N can also be the correct form. But we have to maintain the same order in all the certificates.

In case of existence of your long name, the same thing can be represented diligently with your prudent approach. You can closely examine your long name and ensure that your basic name with which you are being called in your family or in your class should appear first signifying that the same is your basic name and the same is to be followed by your father's name and the last name must indicate your title. If by arranging in that fashion still some parts have not been adjusted, go through the same and see where this could fit ensuring the sequence that the first words are your original name, second being the father's name and rest is the title. The excess terms could fit after your basic name if you scrutinise the same.