In almost all situations, improving your chance to hit is better than improving your damage roll. And thus, Attack Advantage will almost always be preferable to Damage Advantage.
I went ahead and wrote an AnyDice program to compare the two, and if you'd like to go fiddle with it, you can find it here. In the program, I compute average damage per attack for both Attack Advantage, and Damage Advantage.
I ran multiple tests with different weapons, AC bonuses, and Attack Bonuses and came to the following conclusion.
If the target's AC is more than 3 points higher than your Attack Bonus (which is usually the case), then Advantage on Attack Rolls yields a higher average damage than Advantage on Damage Rolls does. This holds up for all weapon dice-sets that exist in the PHB.
So, to give one example: that of a character with 20 STR and a Longsword...
Opposed by an AC of 15, if his Attack Bonus is +11, his Average Damage Per Attack will be...
- Attack Advantage: 9.72
- Damage Advantage: 9.45
If you increase his Attack Bonus to +12, then...
- Attack Advantage: 9.84
- Damage Advantage: 9.99
This pattern holds true as you increase AC...the larger the gap between AC and the Attack Bonus (and, in general, there will be a significant gap between the two) the less useful Damage Advantage will be.
This also follows logically. Advantage on a Damage Roll increases your chances of doing a little more damage. Advantage on an Attack Roll increases your chances of doing any damage at all. So Damage Advantage can mean the difference between doing 6 damage and 8 damage. Attack Advantage can mean the difference between 0 damage and 8 damage.
That being said, I discovered another situation in which Damage Advantage holds up better. If you do not have the two-weapon fighting feature, and so your off-hand damage does not gain the +damage from the attack stat, then the AC/Attack Bonus margin increases to 5, instead of 3. i.e. +10 to hit vs AC 15 with an off-hand weapon (no bonus), Damage Advantage is better. But, even here...it won't be often that you have such a high Attack Bonus against something with an AC that is low enough you only need to roll a 4 to hit it. So, practically speaking...this doesn't matter much.
So, TL;DR: Damage Advantage is almost always inferior to Attack Advantage.
Wikipedia defines the clustering fallacy as:
...the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random.
Basically, a perfectly random die will have long (potentially very long) steaks of seemingly non-random behavior, either very high or very low. Therefore, your observation of low rolls in multiple sessions could simply be true random variation. Streaks, both high and low, might not be common, but you'll definitely notice them a lot more than more varied behavior. After all, a sequence of exactly 19,20,19 will be a lot more noticeable than a 3, 16, 9, even though the exact probability of both sequences is the same.
The only way to truly know if any method or die is biased or not is to do dozens of identical rolls and look at the resulting distribution. Additionally, everything from the die itself to the rolling surface to your throwing technique is going to affect the roll.
How random does it need to be?
Moreover, how random is random enough? If you read the paper that your article is citing, they say that for a 6-sided die that bounces 4-5 times, the probability that the die lands on the face that's the lowest at the beginning is about 20%, whereas the expected random value is about 16%. Would you notice a 4 percentage point difference on a d6 roll?
For a more concrete personal example, I have a gold d20 that that has a very strong bias toward rolling 20s, according to the saltwater flotation method. However, when I use this "golden" d20, I don't actually notice a greater proportion of 20s showing up in my rolls.
Therefore, the answer to your first question is yes: rolling methods do affect the roll, but unless you're using actual loaded dice, the difference is too small to matter. This means that the answer to your second question is also no, as well.
Best Answer
The other answers have made a mathematical error. They neglect the possibility of rolling pairs. You can roll a 6 and a 9 as 6,9 or as 9,6, but there is only one way to roll a pair of 6s. This slightly lowers the probability of rolling equivalent results twice.
The probability of rolling a pair is 1/20 (the probability that the second die lands on the 1 face out of 20 that matches the first), and there is only 1 outcome out of 400 possibilities where the second set of dice match the first if the first is a pair.
The probability of not rolling a pair is 19/20, and there are 2 outcomes out of 400 possibilities where the second set of dice match the first if the first is not a pair.
The probability of rolling 2 sets of 2 dice and getting the same results up to order is thus 1/20*1/400 + 19/20*2/400 = 39/8000, or 0.004875, slightly less than the 1/200 number the other answers give.