How to Make Wudu
Question

Could you please tell me the proper way of making wudu? Is there a difference between men and women in wudu? Can we say “may Allah have mercy on our beloved Prophet”?
Praise be to Allah.Congratulations on your conversion to Islam 
We praise Allah for having guided you and opened your heart. We ask Allah to make us and you steadfast in obeying Him. We appreciate your efforts to learn about your religion and we advise you to strive to acquire knowledge so that your worship will be correct. Try to learn Arabic so that you will be able to read Quran and understand it properly. We ask Allah to bless you with beneficial knowledge. 
How to Make Wudu
With regard to how wudu is to be done, there are two ways; the complete and the acceptable way. Find the details below.
1 – Obligatory parts of wudu, which are as follows: 
1.      Washing the face completely once, which includes rinsing the mouth and nose.
2.      Washing the arms up to the elbows , once.
3.      Wiping the entire head , including the ears.
4.      Washing the feet up to the ankles, once .
What is meant by once in all the above is that the entire part of the body mentioned must be washed thoroughly.
1.      This must be done in order, so one washes the face first, then the arms, then wipes the head, then washes the feet, because the Prophet (peace and blessings of Allah be upon him) did wudu in this order.
2.      This must be done continuously, i.e., the parts of the body must be washed one after the other with no lengthy interruption between washing one part and the next.
These are the obligatory parts of wudu which must be done in order for wudu to be sound. 
The evidence for these obligatory parts of wudu is the verse in which Allah says (interpretation of the meaning): 
“O you who believe! When you intend to offer As-Salah (the prayer), wash your faces and your hands (forearms) up to the elbows, rub (by passing wet hands over) your heads, and (wash) your feet up to the ankles. If you are in a state of Janabah (i.e. after a sexual discharge), purify yourselves (bathe your whole body). But if you are ill or on a journey, or any of you comes after answering the call of nature, or you have been in contact with women (i.e. sexual intercourse), and you find no water, then perform Tayammum with clean earth and rub therewith your faces and hands. Allah does not want to place you in difficulty, but He wants to purify you, and to complete His Favour to you that you may be thankful” [al-Maidah 5:6]
2 – Mustahabb parts of wudu. These were narrated in the Sunnah of the Prophet (peace and blessings of Allah be upon him), the details of which are as follows: 
1.      You must have the intention of purifying yourself and removing impurity. The intention should not be spoken out loud for its place is in the heart. This applies to all acts of worship.
2.      You should say Bismillah .
3.      Then you should wash your hands three times
4.      Then you should rinse your mouth three times, swirling the water around inside your mouth, and rinse your nose three times, blowing the water out and using the left hand to remove the water from your nose.
5.      You should wash your face three times, from the hairline to the jawbone and chin, and from ear to ear. A man should wash the hair of his beard because it is part of the face. If his beard is thin he has to wash it inside and out, and if it is thick and covers the skin, he should wash the surface of it only and run his wet fingers through it.
6.      Then he should wash his arms up to the elbows three times. The arm extends from the fingertips, including the nails, to the lower part of the upper arm. It is essential to remove anything stuck to the hands before washing them, such as dough, mud, paint, etc, that could prevent the water from reaching the skin.
7.      Then after that he should wipe his head and ears once with fresh water, not the water left over from washing his arms. The way in which the head is to be wiped is that you put your wet hands at the front of your head and bring them to the back of your head, then bring them back to the place where you started. Then put your index fingers in your ears and wipe the back of the ears with your thumbs. With regard to a woman’s hair, she should wipe it whether it is loose or braided from the front of her head to the roots of the hair at the nape of her neck, but she does not have to wipe the entire length of her hair.
8.      Then you should wash your feet three times up to the ankles, namely the bones at the bottom of the leg.
The evidence for that is the hadith narrated by Humran the freed slave of ‘Uthman, who said that ‘Uthman ibn ‘Affan (may Allah be pleased with him) called for water to do wudu. He washed his hands three times, then he rinsed his mouth and nose, then he washed his face three times, then he washed his right arm up to the elbow three times, then he washed his left arm likewise. Then he wiped his head, then he washed his right foot up to the ankle three times, then he washed his left foot likewise. Then he said, “I saw the Messenger of Allah (peace and blessings of Allah be upon him) doing wudu as I have done it, then the Messenger of Allah (peace and blessings of Allah be upon him) said, ‘Whoever does wudu as I have done it, then prays two rak’ahs in which he focuses completely on his prayer, his previous sins will be forgiven.’” (Narrated by Muslim, al-Taharah, 331) 
Conditions of wudu
The conditions of wudu are: being Muslim, being of sound mind, having reached the age of discernment and having the intention of doing wudu. Wudu is not valid on the part of a non-Muslim, an insane person, a small child who has not yet reached the age of discernment or one who does not have the intention of doing wudu because his intention is to cool himself down, for example. 
The water must also be pure (tahir), for impure (najis) water cannot be used for wudu. One must also remove anything that could prevent water from reaching the skin and nails, such as nail polish. 
It is prescribed to say Bismillah according to the majority of scholars, but they differ as to whether it is obligatory or Sunnah. If one remembers to say it, it may be said either at the beginning of wudu or during it. 
Difference between men and women in wudu
There is no difference between men and women in the way wudu should be done. 
What to say after wudu?
It is mustahabb to say, when one has completed wudu : “Ashhadu an la ilaha ill-Allah wahdahu la sharika lah, wa ashhadu anna Muhammadan ‘abduhu wa rasuluhu (I bear witness that there is no god except Allah alone with no partner or associate, and I bear witness that Muhammad is the Messenger of Allah),” because the Prophet (peace and blessings of Allah be upon him) said: “There is no one among you who does wudu and does it properly and does it well, then he says, ‘Ashhadu an la ilaha ill-Allah wahdahu la sharika lah, wa ashhadu anna Muhammadan ‘abduhu wa rasuluhu,’ but the gates of Paradise will be opened to him and he may enter through whichever of them he wishes.” (Narrated by Muslim, al-Taharah, 345). A report narrated by al-Tirmidhi adds: “Allahumma ij’alni min al-tawwabina wa’j’alni min al-mutatahhirin  (O Allah, make me one of those who repent and make me one of those who purify themselves).” (al-Taharah, 50; classed as sahih by al-Albani in Sahih Sunan Abi Dawud, no. 48) (See al-Mulakhkhas al-Fiqhi by al-Fawzan, 1/36) 
Can we say “may Allah have mercy on the Prophet”?
 With regard to your saying “may Allah have mercy on the Prophet”, what is prescribed in Islam is to send blessings and peace upon him, as Allah commanded us to do when He said (interpretation of the meaning): 
“Allah sends His Salah (Graces, Honours, Blessings, Mercy) on the Prophet (Muhammad), and also His angels (ask Allah to bless and forgive him). O you who believe! Send your Salah on (ask Allah to bless) him (Muhammad), and (you should) greet (salute) him with the Islamic way of greeting (salutation, i.e. As-Salamu ‘Alaykum)” [al-Ahzab 33:56] 
And Allah knows best.