Your ghusl in the manner described is valid and acceptable, praise be to Allaah. But you have missed out some of the sunnahs, which does not affect the validity of your ghusl.
The reason for that is that ghusl is of two types: acceptable and complete. In the case of acceptable ghusl, it is sufficient to do the obligatory parts of ghusl only, without doing any of the mustahabb or Sunnah actions. He should intend to purify himself, then make water reach every part of his body, using any means whether that is standing beneath a shower or immersing oneself in the sea or in a swimming pool and the like, as well as rinsing the mouth and nose.
Complete ghusl means doing what the Prophet (peace and blessings of Allaah be upon him) did it, with all the sunnahs of ghusl.
Shaykh Muhammad ibn ‘Uthaymeen was asked about how ghusl is done.
He replied: Ghusl may be done in two ways:
1 – By doing the obligatory parts of ghusl only, which is making water reach all parts of the body, and includes rinsing the mouth and nose. If a person makes the water reach all parts of his body, by whatever means, then he has removed the major impurity and purified himself properly, because Allaah says (interpretation of the meaning):
“If you are in a state of Janaaba (i.e. after a sexual discharge), purify yourselves (bathe your whole body)”
2 – The complete way, which is doing ghusl as the Prophet (peace and blessings of Allaah be upon him) did it. When he wanted to do ghusl in the case of janaabah, he would wash his hands, then wash his private part and anywhere else that the impurity had reached, then he would do complete wudoo’, then he would wash his head three times with water, then he would wash the rest of his body. This is how complete ghusl is done.
End quote from Fataawa Arkaan al-Islam, p. 248.
There is no difference between ghusl in the case of janaabah and ghusl following menses except that it is more mustahabb to rub the hair in the case of ghusl following menses than in the case of ghusl following janaabah. It is also mustahabb for the woman to apply perfume to the site of bleeding, to remove any unpleasant smell.
Muslim (332) narrated from ‘Aa’ishah (may Allaah be pleased with her) that Asma’ asked the Prophet (peace and blessings of Allaah be upon him) about ghusl following menses. He said: “Let one of you take her water and lotus leaves and clean herself well, then let her pour water over her head and rub it vigorously so that it will reach the roots of her hair. Then let her pour the water over herself, then take a piece of cloth scented with musk and purify herself.” Asma’ said: How should she purify herself? He said: “Subhaan-Allaah! Purify yourself with it.” ‘Aa’ishah said – as if she whispered it to her – Follow the traces of blood. And she asked him about ghusl in the case of janaabah. He said: “Let her take water and clean herself well – or clean herself thoroughly – then let her pour water over her head and rub it so that it reaches the roots of the hair, then let her pour water over herself.” ‘Aa’ishah said: How good the woman of the Ansaar were! They did not let shyness prevent them from understanding their religion properly.
So the Prophet (peace and blessings of Allaah be upon him) differentiated between ghusl following menses and ghusl following janaabah, with regard to rubbing the hair and applying perfume.
Saying Bismillah when doing ghusl and wudoo’ is mustahabb according to the majority of fuqaha’, and the Hanbalis said that it is obligatory.
Shaykh ibn ‘Uthaymeen (may Allaah have mercy on him) said: Saying Bismillaah is obligatory according to our madhhab, as in the case of wudoo’. There is no text concerning that but they said: It is obligatory in the case of wudoo’ so it is more so in the case of ghusl, because that is a greater purification.
But the correct view is that it is not obligatory in the case of either wudoo’ or ghusl. End quote from al-Sharh al-Mumti’.
Rinsing the mouth and nose is essential in ghusl, as is the view of the Hanafis and Hanbalis.
Al-Nawawi said, discussing the difference of opinion on this point: the views of the scholars on rinsing the mouth and nose are four:
1-That these actions are two of the sunnahs of wudoo’ and ghusl. This is our view (Shaafa’i).
2-That they are obligatory in both wudoo’ and ghusl and are conditions of them being valid. This is the well known view of Ahmad.
3-That they are obligatory in ghusl but not in wudoo’. This is the view of Abu Haneefah and his companions.
4-That rinsing the nose, but not the mouth, is obligatory in wudoo’ and ghusl. This was narrated from Ahmad, and Ibn al-Mundhir said: This is my view.
End quote from al-Majmoo’ (1/400).
The correct view is the second one, which is that rinsing the mouth and nose is obligatory in ghusl, and that is a condition of it being valid.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: Among the scholars are some who say that ghusl is not valid without them (i.e., rinsing the mouth and nose), as is the case with wudoo’.
And it was said that it is valid without them.
The correct view is the former, because Allaah says (interpretation of the meaning): “purify yourselves (bathe your whole body)” [al-Maa'idah 5:6],and this includes the entire body. The inside of the nose and mouth are part of the body which must be purified. Hence the Prophet (peace and blessings of Allaah be upon him) enjoined that in wudoo’ because they are included in the words of Allaah (interpretation of the meaning): “wash your faces” [al-Maa’idah 5:6]. As they are included in washing the face and are included in that which must be purified in wudoo’, they are also included in ghusl, because purification in that case is even more important. End quote from al-Sharh al-Mumti’.
If in the past you did not rinse your mouth and nose when doing ghusl because you did not know the ruling on that or because you were following the opinion of someone who does not regard that as obligatory, then your ghusl is valid and your prayers offered following that ghusl are also valid, and you do not have to repeat them, because of the strong difference of opinion among the scholars concerning the ruling on rinsing the mouth and nose, as stated above.
May Allaah help us all to do that which He loves and which pleases him.
And Allaah knows best.