All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Washing one's hair is a mandatory act in the Ghusl for Janaabah (ceremonial washing which is performed following sexual intercourse, ejaculation, menstruation and post-delivery impurities). Similarly, wiping on one's hair is an obligatory act in Wudu (ablution). So, neither washing the hair in Ghusl, nor wiping it in Wudu can be left out regardless of the length of the hair the person has and regardless of its type: wavy, curly, kinky, …etc. So, any Wudu that is done without wiping or any Ghusl done without washing hair are null and the impurity is not lifted in this case. This leads also to the invalidity of any prayer performed following a non-valid Wudu or Ghusl.
The Prophet said: “No prayer is valid without purity.”
So, one must adhere to that which is obligatory in order to fulfill what he is supposed to perform in the correct way. On the other hand, wiping on the hair in Wudu does not cause any falling out of the hair as it should be slight because it is different from Ghusl.
As for Ghusl, if it is the main cause of the falling out of hair which might lead to complete baldness, then it is lawful to wipe on it instead of washing it. This was stated in a previous Fatwa issued by Islamweb. Nevertheless, if the falling out of hair is caused by drying it up, combing it, styling it; etc.; then one should stay away from these reasons and not resort to wiping on the hair. In this case, one should perform full Ghusl. You should also know that it is the doctor who defines the real cause of the falling out of the hair and it is he who gives the right advice which enables the Mufti to issue a sound Fatwa.
As you know, washing the feet is an obligation in Wudu and Wudu is invalid without it unless the person is unable to wash them due to an illness that gets worse or which is stagnant; in this case, the washing can be left and the person performing Wudu does his normal Wudu and when he/she reaches the feet, he/she performs Tayamum (dry ablution) as a substitution to washing the feet.
Finally, you may use shoes or socks whose ruling is made clear in Fatwa 82001.
For more benefit on obligatory acts in ablution, and which are detailed in sequence order, please refer to Fatwa 326854.
Allah knows best.