Regarding the Ice Climbers being like two characters in one, does Nana the pink climber have a fixed damage percentage before being able to get knocked out? If she manages to get knocked out, What are the draw backs the player will suffer besides not able to do a jump recovery to its maximum extent?
I think I'll take a stab at making a write up for this one. If anyone has anything to add or disagrees with me, by all means step in because a lot of this is going to be opinion based.
To get the first question out of the way, there is no set fixed damage percentage for Nana to get knocked out. She takes damage like any other player and hit knockback increases accordingly. That being said, for the most part her cpu is programmed to automatically return to Popo. Couple that fact with her limited recovery, she is fairly susceptible to being gimped at low percentages.
Now on to your more interesting question, in my opinion.
From a metagame stand point, you lose the following in Solo Popo (Sopo):
1. Recovery options - As you mentioned, Sopo suffers from severely limited recovery. Belay has little to no range and Nana is no longer present to serve as an attacking device. Squall hammer, at best, will only recover horizontal distance, which is much worse than the vertical distance one gets with a synced squall.
2. Damage Dealing Capabilities- Obviously, when Popo and Nana are attacking simultaneously, they do more damage. Two hits are better than one. Because it requires more hits to rack up damage, Sopo is also hampered by staling moves.
3. Desynced game - In contrast to above, rather than doing two players worth of damage at once, desyncs allows a player to use one ice climber to cover the moves of another ice climber, effectively reducing move lag time and punishing opportunities for opponents. While technically not desynced, a blizzard loses the rear cover Nana provides as well.
4. Infinite Chain Grabs and Hobbling - Much of the IC metagame is to punish opponents mistakes by taking a stock, a trait that really isn't rivaled to the same extent by any other character.
5. Nana Meat Shield Factor - Especially in projectile based opponents, having Nana around might reduce the registered percentage of damage a player takes, thereby giving an IC player an advantage to win a round when a percentage tiebreaker comes into effect.
6. Lack of finishers - Outside of a smash attack, Sopo really has no finishing moves. Bair/Uair will probably be too stale from racking up damage, and it's harder to get a full charged smash attack to land without Nana holding the opponent. Also, Popo's Fair does not meteor like Nana's. Bungie-Jumping as an edge guarding technique is now impossible, so the IC's lose the longest ledge hogging invincibility time in the game.
However, more important, are the mindgames that one loses when forced to only go Sopo. Outside of the previously mentioned metagame disadvantages, a Sopo suffers from the following:
1. Lost desync deception: A player's ability to confuse the opponent with activity is effectively cut in half. Especially if a player effectively uses Nana's smash charge dragging capabilities,
2. Loss of enemies tentativeness to approach - Let's face it. One of the huge advantages an IC player has is the fact that the game really slows down for us because opponents are so scared of making a mistake and losing their stock via punishing infinte chaingrab. When Nana dies, they can breathe a sigh of relief because our main punishing weapon is gone from the match. They can afford to make a small mistake because they won't lose their stock for it.
Remember Popo's prayer: Though I walk through the valley of the shadow of death, I will fear no evil, for Nana is with me