Hair

Yes it is allowed to wear such wig as far as it does not obstruct Wudhu or Ghusl.

Wassalam.

Yes it is permissible to cut the hairs around the eyebrows as far as it does not make the male look like a female.

Wassalam.

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. 
Wassalam.

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.

Wassalam.

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.

Wassalam.

First of all, we as Muslims believe that Allah is The All Knowing, Most Wise, Most Merciful and we believe that everything comes from Him is based on full wisdom for our interest, that is why we never question His Wisdom in any matter. We believe that the full reason is best known by Allah (SWT), the Messenger (SAWA) and Ahlul Bayt (AS) who are the real successors of the Prophet (SAWA).

We also believe that our human abilities of thinking are limited and can never reach to everything. We know for sure that every order from Allah (SWT) is based on our benefit and we don't question the details.

Hijab is obligatory on females because Allah (SWT) who created males and females made males more attracted to females than females to males in usual situations. Hair of man can be attractive to some females in some situations but females are in general are much more attractive to males. This can be one of the reasons. Men are responsible to work hard to earn livelihood for their families. Hijab of women is for maintaining her dignity, modesty, safety and honour. Hijab also provide safe atmosphere away from sinful acts. Allah (SWT) says in Quran about Hijab on women (So that they are known so that they should not be harmed ( كي يُعرَفنَ فلا يؤذَين)) 33:59).

Wassalam.

Removing such hair and hair on the pubes and under arms is required because it helps keeping the body clean, tidy and hygienic. Such unwanted hair should be removed at least once in maximum of forty days. Leaving it more than forty days in disliked (Makrouh). Leaving unwanted hair for longer period might cause harm for health because of dirt which can accumulate on it and can also disrupt cleaning the area.

'Wassalam.