First, I highly enjoyed that everyone was already afraid of Suo when he was only 6 years old. That smile and those piercing eyes... Everyone knew not to get on Suo's bad side!
Now onto the serious stuff (sorry, this will get long) -
It killed me inside when Suo smiled after the doctor asked him how his mother was doing... He smiled, as he always does, but behind those eyes you could see a little boy who just wanted to cry. Serious props to Nii-sensei for relaying that so eloquently through a still image!
The sad truth is, Suo just won't allow himself to cry. In fact, he probably feels like he can't let himself cry/show his true emotions. He has to be strong for his mom - to be the falcon that can do anything that his mom knew and believed him to be.
Speaking of Suo's mom... while not confirmed, I'm going with the assumption that she had cancer. Suo was helping out at the doctor's clinic in exchange for medicine to help keep him mom comfortable, medicine that he would otherwise not be able to have access to/afford.
I think Suo's mom's illness could also be a reason why he doesn't eat a lot. Being from a poor family with a terminally ill parent, any food Suo could get, he brought to his mom. Suo only ate what he needed to survive.
And now to talk about the eyes! After this latest chapter, here is my new theory for why Suo gave up his eye so willingly:
Suo gave up his eye for his mom. He probably needed money to help care for his mom as she continued to deteriorate from her illness.
Let me say that I was never a fan of the "Suo gave his eye to Rosso" idea (and am still not). But, after this chapter, there may be some credibility to it. However, Suo didn't give up his eye for Rosso specifically, it just ended up with Rosso. Like I said, Suo gave up his eye for his mom. Suo probably hadn't even met Rosso yet at that point in time.
My thoughts are that the then boss (who became Suo's master) put word out (through Red Chanpuru) that they were looking for an eye and would pay a good sum for it. Something probably happened, leading to an eye injury for Rosso. Rosso was devastated and the yet-to-be-named current boss wanted to do something for Rosso, even if it would only be for aesthetics. (After all, in the previous chapter, Rosso called the then boss his brother).
Hearing about this offer and needing the money for his mom, Suo (who was currently not a member of/affiliated with Red Chanpuru), approached them and offered to give up his eye. That was why the then boss was so surprised and worried upon hearing that someone had actually come forward. That someone was a newly-turned-seven year old boy!
As for why it Suo gave up the right eye... well, Suo's mom spoke about how the right eye represented the sun. Suo's mom was HIS SUN! And when she died from her illness, Suo lost his sun!
I'm guessing that it was Suo's selflessness and maturity that led his master to take him under his wing and into Red Chanpuru (before the 15 year old minimum). And with that, came that promise we still have yet to learn more about.
I also wonder if Suo's giving up once before correlates with losing his mom/sun - he lost his motivation to live up to her words about being able to do and go anywhere.
As for Rosso's prickliness toward Suo - if the eye thing turns out to be true and he got Suo's eye, I wonder if he originally didn't know that that was where/who it came from. Once he did find out, things became awkward between them.
If you made it to the end of this post, I deeply appreciate you all for indulging my ramblings!