Can the nikah be valid if the bride's mother is present and the two witnesses are online on Skype or not present?
Please specify for both conditions and also tell is it is necessary to fill a Nikkah nama form for the marriage.
Initially, I ought to point out that (in accordance with the viewpoint of Grand Ayatollah Makarem Shirazi (as a Marja' al Taqlid of Shia Islam):
Two parties can perform the concubine or Siqah and there is no necessary for witness. (and actually there is no difference between being present or by phone...). But the significant point is that the presence of girl’s father is necessary.
Or according to the viewpoint of Grand Ayatollah Khamenei (as a famous Marja'al-taqlid of Shia Islam):
from the viewpoint of Sharia, having the witness is not as a reason to show its accuracy, and the Nikah (Aqd or) without witness is true. But the presence of witness is necessary for Talaq (divorce).
Another issue that you ought to pay attention to, is that:
According to Islam (all Maraja’ al Taqlid(of Shia)), a girl who has reached to the age of puberty(bulugh), in truth, if she can recognize her benefit(Masleha), in case of being Bakereh(virgin), ought to obtain the permission of her father (or her paternal grandfather).
And filling the Nikkah-Nama is another point which is related to the rule not Sharia, otherwise this is not necessary or obligatory for you to fill it (according to the Sharia).
The bride's mother cannot be wali (because wali should me a man; but i do not yet know proofs for that wali should be a man).
The Messenger of Allah said: "There is no marriage except with a Wali."
Aishah narrated that: The Messenger of Allah said: "Whichever woman married without the permission of her Wali her marriage is invalid, her marriage is invalid, her marriage is invalid. If he entered into her, then the mahr is for her in lieu of what he enjoyed from her private part. If they disagree, then the Sultan is the Wali for one who has no Wali."
I think skype is not allowed because it can be faked. But, maybe, it is OK in some cases, like if there is groupchat with high quality video and audio and all people already know each other and see that connection through skype is ok, and wide angle or panoramic cameras are used to see that all is OK.
Nikah-nama is not required but announcing the nikah is required: Sunan Ibn Majah, Vol. 3, Book 9, Hadith 1896:
It was narrated from Muhammad bin Hatib that the Messenger of Allah (pbuh) said: “What differentiates between the lawful and the unlawful is (beating) the Daff and raising the voices (in song) at the time of marriage.”