Why did Takechi decided to kill Yataro's partner?
this is one of the controversial areas of this episode. Takechi decided to kill that man because Ryoma told him that someone sent the police (yataro and that man) to catch the person who killed Toyo and Takechi was their suspects. But many (including me) don't understand this part because Ryoma knows that he himself is their suspects rather than Takechi, so it seems strange why he told Takechi the police was chasing him. Or maybe Ryoma knows Takechi is the one who ordered to kill Toyo, and as a friend, he warned Takechi about this. But somehow his warning caused the death of one more person, and which seems to be what he has expected (he asked Yataro to go back to Tosa immediately because it is too dangerous to stay in osaka, this sounds like ryoma knows Takechi will be going to kill the police) Edit: just saw a post in 2ch also discussing this question. The person said maybe ryoma believed that the one sending the police think ryoma was instructed by Takechi to kill Toyo. As he was now being chased, that means Takechi was also in danger. Ryoma did not know in fact, he is the only suspected person.

Who is the sickly young man that Takechi bows to?
About the sick young man, he is the leader of Tosa (succeeding his father's position). At that time there was a campaign (?) called Jyouraku (上洛), which means going to Kyoto. The leaders of the province (Han) go to gather at Kyoto to show their support to their King in Kyoto and also as a support to their King's will of expelling the foreigners. The Tosa young leader also joined this Jyouraku (Takechi and another old man persuaded him to join). During the trip he got sick and was forced to stop at Osaka, and that's why Takechi was very worried (they were left behind by other provinces)

More discussion and theories on Takechi's actions
I think Takechi killed the guy to frame Ryoma. Since everyone knows the Tosa policeman is after him and they had a fight in the market. If he dies => Ryoma is the only suspect => Ryoma must have killed Toyo, otherwise he wouldn't need to kill the police.

In the meantime, Ryoma in his true goodness, still wants to pull Takechi back from the brink. He probably also wants to know if Takechi is being suspected too. That's the only reason I could think of, why he would want to talk to Takechi ........ in the Tosa clan lodgings (where the Yamauchi boy and Takechi were staying) .......... which btw is also full of guards protecting the frigging daimyo, and where he's to be killed if caught for desertion alone, not to mention assassination of the under-secretary. Where's a more dangerous place to go, other than back to Tosa?

R: Oi, Takechi-san, the police are here to investigate me. Since we're so close, they might check you up too, also as you're the one who stands to gain the most out of Toyo's death. Please be careful. BTW I know you killed Toyo. So please don't go in any further. Please don't kill anyone else who stands in your way anymore!

But Ryoma's not naive anymore, he's probably knows the flipside in case Takechi's really gone past the point of sanity and refuses to listen to him. But he's got to try anyway. Now he knows how far gone Takechi is.

At the same time, Takechi wants to 'look' the good man (aka Senator Palpatine), so he won't snitch on Ryoma when they met either, or people will know he's the snitch (which will bring out speculations on why he'd turn his good friend in) and Ryoma won't be useful anymore as the continuing scapegoat in future.

There could be another way of looking at it, Ryoma could have talked to Takechi just to test him, but I don't want to believe that Ryoma would purposely put the life of another man at stake, just to check if Takechi has really gone evil. After all, as someone in the d-addicts forum said, this Ryoma's too full of goodness.

A counter argument
Yes, i agree that ryoma talked to takechi just because he wants to give him some warning: he is now being suspected so he shouldn't do anything wrong again. But somehow he does not know that takechi has already become an evil and when receiving this warning, he decided to kill one more person instead of stopping doing wrong thing. And that's why in the last shot ryoma looks so sad when he saw the dead body, he is sad not only because someone dies, but also because he realises his old friend has totally become another person.

But i think Takechi killed the police not because he wants to frame Ryoma, he just wants to clear the obstacle in front of him. He believes what ryoma said to him, i.e. he is the suspect, and he and ryoma are regarded as in the same group in the eyes of others, so if ryoma is suspected, he is also in danger. So i think he killed the police just because he thinks the police is his obstacle