Are husband and wife allowed to dance with each other alone in their room?

Dancing is prohibited.

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My wife wants to know if it is permissible to trim the eyebrows with a machine instead of plucking which is forbidden.

It is not permissible to trim one's eyebrows in any way, by using the machine or by plucking.

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Can husband and wife play wrestling?

Does a husband and wife really play wrestling?

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Is it permissible for women to cut their hair if there is fear that split ends might be coming soon?

A woman may use medication to avoid split ends.

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Should I allow my wife to dance on her brothers wedding if its just a small dance in presence of females only. Shes really insisting.

You should not permit your wife to dance even though she is insisting.