Why kill Takagi?

Because Hans was smart and you are not really thinking.

Think about it. If he did not kill Takagi, what is going to happen next?

Hans would have to put him back to the group because he did not have enough men to guard him separately. Takagi would be under the impression these people did not dare to kill. So let's plan a revolt.

Takagi was only valuable to get the code, and it's quite clear there is no way to get it out of him. So killing him was the logical thing to do.

The best reason I can think of is that he has told Takagi his secret plan, he knows they’re robbers not terrorists, so once it’s clear that Takagi doesn’t know the code he’s both useless plus he’s a potential liability if he were to somehow reveal the plan to others.

Blowing his head off is rather… extreme and over cautious, but I can sort of see the sense.

Mr. Takagi knew the key because the key is shown to the audience on Theo's computer monitor as "Red Castle", and visual cues are given showing "Red Castle's" derivation from "Akagi". This seems too coincidental for me.

Why did Mr. Takagi choose to lie to Hans? Mr. Takagi likely suspected that the lives of himself and his employees were forfeit the minute the terrorists got their hands on the vault, and he viewed his life as a necessary sacrifice to buy time.

This was the smartest move he could make, as the terrorists were planning on killing everyone by blowing the roof. Ultimately his sacrifice was in vain as the terrorists' plan was occupied by FBI intervention, for them to turn off the power.

because all things pass away in the end

He should have brought other hostages in and be like “give me the code or I’ll blow this persons head off”

My take...

The plan was always to kill everyone anyway. But they had to maintain the pretext of being terrorists, and they needed to keep the hostages alive until they had the bonds in hand. The first reason to keep the hostages alive til then is that it was all part of the charade. The second reason was practical... if their plan went south, they still had bargaining chips to arrange a getaway.

So there were two ways that the plan could have unfolded.

i. Takagi knows the code and gives it to them. They get into the vault sooner, get the bonds and move up the timing of the plan.

ii. Takagi doesn't know -- or won't give them -- the code. They revert to Plan B which involved using the FBI to get into the vault.

In either plan, Takagi had to be killed in a public way. Primarily, his execution was necessary to keep the hostages in line through fear. Hans could also use his execution as part of the charade for the benefit of the outside authorities... it demonstrated to them that this 'terrorist' group was legit. They could even have used a terrified hostage on the phone to describe the execution to enhance the whole 'terrorist act'.

So, in the end, they had always planned to kill Takagi in advance . They just decided to turn it into an advantage they could use by speeding up the timetable if they could extract the code from him.

There's always two ways to look at these things. There's a real-world vantage and there's the movie world. This Takagi killing was a plot artifice designed to create clock-ticking tension and for the later money shot of the safe finally opening up. From a real-world perspective it was a dumb move by Hans. Why not at least try torturing or executing an employee or two to motivate Takagi giving them the combination? They'd already murdered the guy at the front entrance, so it's not like they were worried about having a nagging conscience.