Why is it placed on our head after Darsan?
If I am correct, the Satagopuram is not offered in Shakthi temples. What is the reason behind this? (In which kind of temples can this be seen?)
Do we need to chant any mantra/hymns while Satagopuram is placed on our head or does the priest chant any hymns?
Can it be placed on all of us unconditional of our purity or are there any exemptions?
It's because the Sadagopuram, AKA Sadagopam or Sadari, is unique to Vishnu temples.
The Alwars (also spelled Azhwars) are a group of 12 ancient Vaishnava saints who lived in Tamil Nadu and are famous for their poetry in praise of Vishnu, which was pivotal in the development of the Sri Vaishnava sect (of which I'm a member). One of the most famous Alwars was called Nammalwar or Sadagopan, and he spent his life under a Tamarind tree in the town of Kurugur, but he was still able to compose poems about many of the Divya Desams, the 108 sacred places of Vishnu praised in the Alwars' poetry.
In one of his poems praising the deity of the famous Uppillaiappan Vishnu temple, Nammalwar says this (see page 1): "I, Sadagopan of Kurugur [am] one who is ... joined with His Holy Feet that grew to the heavens." The "grew to the heavens" part is about Vishnu's incarnation as Vamana the dwarf, but what's important for our purposes is that Nammalwar said that he was one with Vishnu's lotus feet.
So Sri Vaishnavas started using a crown, called the Sadagopam, to represent the head of Nammalwar (since his name was Sadagopan), and on the top of the Sadagopam are the footprints of Vishnu, as you can see in the image above. The idea is that just as Nammalwar surrendered to the lotus feet of Vishnu, so should you. That is the origin of placing the Sadagopam on people's heads in Sri Vaishnava temples.
@KeshavSrinivasan has already given a fantastic answer on the origin of the practice, but I would like to address the purity aspect which you brought up.
Everyone can in fact receive the Sadari, but there are differences in the way it is presented. Most people who arrive at the temple in everyday clothes will receive the Sadari placed once on their head (note that there are not any mantras recited by anyone during this process).
However, it is different for those who arrive in Swarupam or Panandathiruman like this:
The placement of 12 thiruman on the body may be done after taking part in the ritual of Samasrayanam which includes the branding of the Sanga and Chakram, or conch-shell and discus, on the shoulders. Samasrayanam represents initiation into Sri Vaishnavism. This ritual can be performed by an acharya on anyone, regardless of caste.
If you arrive at a temple in this manner, the Sadari is placed first on the head, then on the right and left shoulders at the spot of the brandings, and then once again on the head. In some temples they might shorthand it by simply placing the Sadari twice on the head, or they might extend it by additionally placing it on the praying hands before it returns to the head.
You can think of it as an extra reward for those who have undertaken this extra step of commitment towards Perumal.
