The Prince's Ancestors

Part A:
Once upon a time, there lived a fine young prince. He was next in line to the throne in a dynasty that stretched back for centuries. He had the official records that showed that his father's father's father's... 20 generations back had effectively created a country by unifying warring tribes.

Though it should be noted that his male line ancestors were not always very good husbands, and indeed their wives sometimes had lovers when the king was away fighting wars.

At each generation is was estimated that there was a 10% chance that the father's name on the birth certificate was not in fact the biological father.

What is probability that the man that the prince thinks is his father's father's father ... 20 generations back on the male line is indeed his true biological male line ancestor?

Background maths level required: high-school.
Difficulty level: fairly straight-forward.

Part B: 
Over the course of every generation, immigration caused a 10% population increase.

The true figures for the number of biological children each person had are as follows:
20% of people had no children that survived
20% of people had 1 child
20% of people had 2 children
20% of people had 3 children
20% of people had 4 children

Assume that when people paired off to reproduce, their partner was effectively just randomly chosen from their generation of population.

At the time of the founding king, his generation accounted for 1 million people, though the full population of the country was higher if we include the older people who were alive at the time.

What is the probability that the founding king was indeed an ancestor of the young prince?
Please give your answer as a percentage with two digits after the decimal point.

Level of maths required: some basic university maths is probably required.
Level of difficulty: for someone who can write a bit of code, fairly straight-forward