Computing the ricochet rate is actually simpler than it looks:

• The chance of 2d8 NOT rolling doubles is (1*7/8)=7/8=87.5%
• If we get to choose whether to reroll one of the two dice we multiply that AGAIN by the chance of rolling doubles, giving us 7/8*7/8=49/64=~76.6%
• On a critical hit (5% of the time), you roll a total of 4d8, remembering that ANY double results in a ricochet.
• The chance of 4d8 NOT rolling doubles is (1*7/8*6/8*5/8)=41%
• If we get to choose whether to reroll one of the four dice we multiply that AGAIN by the chance of rolling doubles on the fourth die, giving us 41*62.5%= 25.6%
• The upshot is that a standard chaos bolt will ricochet 12.5% of the time, but an empowered chaos bolt will ricochet 23.4% of the time.
• On a critical hit, a standard chaos bolt will ricochet 59% of the time, while an empowered chaos bolt will ricochet at a whopping 74.4% rate!

In order to figure out HOW MANY times we'd ricochet on average, we use a version of the multiplier effect. Simply put, our average number of targets hit is 1/(chance of no ricochet).

• For a regular bolt, that's 1/(7/8)=8/7=~1.14, a twinned spell bolt multiplies this by 2 for 2.28
• For an empowered bolt with one reroll, that's 1/(49/64)=64/49=~1.31, 2.62 as a twinned spell
• For a crit from a regular bolt, that's 1/(.41)=2.44 targets, 4.88 as a twinned spell
• For an empowered crit, our average is 1/(.266)=3.9 targets, 7.8 as a twinned spell.

Assuming every attempted attack hits, your expected number of targets including the 5% chance of a crit is:

• (1.14*95%+2.44*5%)=1.21 for a regular chaos bolt
• (1.21*2)=2.42 for a twinned chaos bolt
• (1.31*95%+3.9*5%)=1.44 for an empowered chaos bolt
• (1.44*2)=2.88 for a twinned, empowered chaos bolt.

For our first two results, the expected damage is the number of targets multiplied by the average damage roll of 2d8 (which is 9). So a standard bolt deals 10.89 damage while a twinned bolt deals 21.78 on average.

Figuring out the average damage on the empowered bolt is... more complex, since you wouldn't reroll when your initial rolls were doubles, you have to compute the probability of a rerolled die exceeding the original damage die, and recompute the average value for the rerolled die.

For the case of an empowered hit from a chaos bolt, we can divide our calculation into three parts:

1. The expected value of rolling doubles
2. The expected value of either the die dropped or the die kept in the event doubles are not rolled
3. The expected value of the replacement die roll

Note that we don't reroll doubles, nor is there never a point where we don't reroll without them - we are optimizing for ricochet and thus will ALWAYS go for doubles.

1. This is fairly straightforward, since the probability of rolling doubles is 1/8 and the probability of rolling any particular set of doubles is 1/8th of that (1/64). The sum of (1+1)/64 + ... (8+8)/64 is identical to 2 times the average roll ((1+2+3+4+5+6+7+8)/8) times the probability of rolling doubles (1/8). The result is an EV of 9/8 for the doubles.
2. This gets a little bit more complicated. We need to compute the probability for each possible number that it will remain when the other die is dropped.
   • There is no chance for us to have a 1 remaining when dropping a die, since that would mean the other die was also a 1 and we would not choose to reroll.
   • For a 2, we could have rolled (2,1) or (1,2) out of 64 options, so we have a 1/32 chance that our highest die before reroll is a 2.
   • The probability formula for having a highest side k on two n sided dice is 2*(k-1)/n^2. Note that summing this formula across all sides and the probability of rolling doubles adds to 1, since you will either have doubles OR the remaining die.
   • The expected value of the die remaining after drop is 5.25
3. For the reroll, our expected value is the average roll (9/2) times the probability of not rolling doubles (7/8), which is 63/16, about 3.94

Adding all of these together, we get an EV of 10.31 for an empowered Chaos Bolt, a 15% increase in damage. But what about a critical hit? Computing the "lowest die" probability is a significantly more complicated problem that I'm still working on...

Placeholder until I can craft a computational model to explain the last part of the question.

On a hit, you deal damage

The spell states that, "on a hit, the target takes 2d8 + 1d6 damage". Nothing later in the spell contradicts this, so no matter what you roll on the dice, the target takes damage. Then you might make another attack roll etc., but that doesn't change the initial attack.

If the Bolt only bounces and doesn't deal damage, it is strictly inferior to Chromatic Orb

Chromatic Orb is another 1st level spell that Sorcerers get access to. It does 3d8 damage of the type you choose. If Chaos Bolt didn't do damage on a bounce, it would do less damage, have an uncontrollable type, and target unpredictably, making it worse in every way to Chromatic Orb. The fact that it can chain and do more damage to more targets is what makes it viable (and fun!).

The 5e developers generally don't make options that completely invalidate other options, and they wouldn't introduce one like Chaos Bolt in a supplemental book when Chromatic Orb was already in the PHB. They certainly make mistakes from time to time but this seems like an obvious one, and the fact that an un-chaining Chaos Bolt would be useless adds weight to the reasoning I used above.