Summarized Fatawaa

Is it permissible for a woman who is in her confinement to fast upon seeing blood signs that are neither black, nor red ?

Brownish, reddish, and yellowish discharges are all considered confinement until there is no colored discharge which marks the end of confinement. Moreover, the aforesaid woman is prohibited from fasting until she attains ritual purity. And Allah Knows Best.

What is the ruling of Sharia on uncovering Awrah  before  a female laser-hair-removal specialist to remove thick hair knowing that I`am engaged and feel embarrassed and this treatment is guaranteed and definitive?

Praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
It is lawful for a Muslim woman to look at the body of another Muslim woman except what is between the navel and the knees.  In addition, it is lawful to look at the whole body for treatment; however, having thick hair isn`t considered an illness. And Allah The Almighty Knows Best.

Is it permissible to make ablution without answering the call of nature after waking up?

Washing front and back openings (Istinjaa`) is not a condition for the validity of ablution because it is done to remove Najaasah (impurity), thus if there is no Najaasah, there is no need for that, and then it is permissible to make ablution without answering the call of nature. However, prayer is invalidated if there is Najaasah on the anus ,or the front opening, so it (Najaasah) must be removed for the prayer to become valid, and not the ablution.

Should water be put in the deceased`s mouth and nose while washing him/her?

Putting water in the mouth and nose of the deceased while washing him is impermissible.