Hair is a protein filament that grows from follicles found in the dermis. Hair is one of the defining characteristics of mammals.
The human body, apart from areas of glabrous skin, is covered in follicles which produce thick terminal and fine vellus hair.
Wearing this type of hair for men or real hair extensions women aren't haram but if it was from some non-Muslims who are not Ahlul kitab, then it would be najis according to many scholars.
May Allah grant you success
Yes it is permissible to cut the hair between eyebrows.
Real Hijab requires the woman to cover her hair and body with out showing its shape. Wearing her hair in bun goes against this condition of Hijab.
Yes it is allowed to wear such wig as far as it does not obstruct Wudhu or Ghusl.
In fact, it is a very good thing. Although you might think that you are distracted, or your mind is somewhere else, but getting used to remembering Almighty God frequently is a wonderful quality.
It will minimise the chance of committing sin, and also keeping you vigilant and alert. Gradually, while multi-tasking, you will be able to master your concentration, and it will all have positive impact as well.
With prayers for your success.
Yes it is permissible to cut the hairs around the eyebrows as far as it does not make the male look like a female.
The place of Mas-h مسح in Wudhu must be dry before wiping over it. Your other hand is making it wet before wiping which is wrong.
Yes, it is allowed to cut nails on a Tuesday.
For further information about cutting of nails, please refer to this Q&A:
And Allah knows best
Neither washing nose is obligatory in Wudhu nor washing mouth. Both of them are just recommended and not obligatory. Washing head or feet in Wudhu is wrong because you should wipe over them and not wash them.
If you are sure that your Wudhu was invalid, you need to perform it properly and re do the prayers which were done with wrong Wudhu.
You will not be able to do both wudhu' and ghusl simultaneously and at once.
So, you can either do wudhu' before ghusl, or after ghusl.
When you wipe your head during wudhu', you can use four fingers, or three fingers. Both cases are fine.
And Allah knows best.
Synthetic wigs are allowed whether made out of human hair or out of any other material. The point is when the wigs are planted in the skin, such wigs prevent water from reaching to the skin and makes a problem in the Wudhu. If wigs are removed before performing Wudhu then there is no problem at all. Wigs can be even worn after performing Wudhu while performing Salah Namaz if the wigs are Tahir. In case Wigs are made of human hair from Najis source like Mushriks then Salah with such Najis hair will be invalid.