Did Shiva drink the entire poison?

During the churning of the ocean by devas and asuras to obtain amrita, a very powerful poison came out.
This was called Halahala or Kalakoota.
The term Kalakoota means that even Kala(Shiva) fainted after drinking it.

What does Halahala stand for?

It represents every form of evil that exists in the world. 

When the poison came out, what exactly happened?

Vishnu instructed Vayu to take hold of the poison.
Vayu then pulverized the poison and gave a small portion to Shiva.
It was consumed by Shiva.
As a result, he became Neelakantha (one with a blue throat).

How was Shiva able to digest the poison?

Because he took it after meditating on Vishnu and chanting Namatraya Mantra.
Shiva himself tells Parvathy about this in Uttara Khanda.232.18 of the Padma Purana.

एकाग्रमनसा ध्यात्वा सर्वदुःखहरं प्रभुम् ।
नामत्रयं महामन्त्रं जप्त्वा भक्त्या समन्वितः ।
तद्विषं पीतवान् घोरमाद्यं सर्वभयङ्करम् ।
नामत्रयप्रभावाच्च विष्णोः सर्वगतस्य वै ।
विषं तदभवज्जीर्णं लोकसंहारकारकम् ॥

The Kaliyuga is evil, but why?

Shiva took the poison in his palm from Vayu.
After drinking, the small particles that stuck to Shiva's palm spread throughout the world and became the evil of Kaliyuga.

What did Vayu do with the whole poison?

He consumed and digested it.

This incident is mentioned in Rigveda.10.136.7.

वायुरस्मा उपामन्थत्पिनष्टि स्मा कुनन्नमा ।
केशी विषस्य पात्रेण यद्रुद्रेणापिबत्सह ॥

Vayu's role in neutralizing the poison is also mentioned in Mahabharata Shanti Parva 351.26-27.

....नीलकण्ठत्वमुपगतः ॥
अमृतोत्पादने पुनर्भक्षणेन वायुसमीकृतस्य विषस्य...

Bhimasena's role in suppressing the evil of Kaliyuga

Bhimasena, Vayu's son is prayed to during Salagrama Puja that he may destroy the evil of Kaliyuga with his gada.

As you can imagine, only a few particles of the original poison have spread around the world, and they are causing so many problems.
How kind our Gods have been to us!
They allowed only a little bit of the poison into the world.