Solution :

1.

Stage1: (Common Sense)

a. Copilot and a Passenger, who is a top mathematician (say X), live in the same city because "If two persons play the same poker games, it is only reasonable to deduce that they live in the same city".

Stage 2: (Elimination)

a. In the last premise, Smith refers to crew member Smith. So, Smith is not the navigator because "A person can't beat himself in any game".

b. Robinson is not the copilot because "Passenger Robinson lives in Los Angeles, while, the Passenger with the same name as the copilot lives in Chicago".

Stage 3: (The opening link)

a. X is Smith because

i. X lives in Omaha because the copilot and X live in the same city, and the copilot lives in Omaha.

ii. X cannot be Passenger Robinson because Passenger Robinson lives in Los Angeles but  X lives in Omaha

iii. Also X cannot be Passenger Jones because X is a top mathematician while Passenger Jones forgot the math he ever knew.

b. Passenger Jones lives in Chicago because Passenger Smith (X) lives in Omaha and Passenger Robinson lives in Los Angeles and one among the Passengers  should live in Chicago.

c. Jones is the copilot because the Passenger with the same name as the copilot lives in Chicago and the Passenger who lives in Chicago is Jones. 

Stage 4: (That one link, unfolding the others)

a. Robinson is the navigator because Jones is the copilot and Smith is not the navigator.

b. Smith is the pilot because Robinson is the navigator and Jones is the copilot.

Therefore, the answer is

Smith - Pilot

Jones - Copilot

Robinson - Navigator

See, how much thought have gone into making the final deduction. Our ancestors understood the whole thought process that takes place in the brain leading to any deduction. This certainly helped them in their acquisition of knowledge too. Tarkam is an exposition of this fact. 

Now, try to find the paksha, hetu, vyaapti and saadhya in the above arguments as an exercise.

For the remaining puzzles, only hints to the answers are given. Give some food to your brain by figuring out the rigorous solutions yourself. Find the paksha, hetu, saadhya and vyaapti too.

2.

The color of the first person's hat is red. The result is actually got by  'thinking about thinking'. The first person inferred what thought went on in the heads of the other two, before they confessed that they didn't know the color of their hats.

3.

The second native is a North-sider while the third is a South-sider. This answer holds good, irrespective of the tribe to which the first native belongs.