Why is spade so indifferent when hearing his partner is killed?

Because his partner was such an awful actor, that's why. When he was shot he closed his eyes as if he was dead fast-as-a-bullet before he begun falling down. Decades before this movie cowboys at least managed to make an Argh! a twist... and then hit the ground. This guy seemed to have been instructed "Go in and do everything quick. Put a lecherous face at the office and die real jiffy. You'll get two bucks and a nickel and a cup of coffee. Action!"

My impression was that they just business Parters and nothing less. A very professional relationship and that's it. Not friends at all
Oh GOOD!,my dog found the chainsaw (Edit God I hate going back and discovering Typos)

I think the uncertainty about Sam's character is also a device to keep a novice audience guessing what Sam will do in the end. Even after his lengthy dialog with Brigid, Sam's "Here's another one for you" is still constructed to be the film's "surprise" dramatic climax. [Did that last kiss change his mind?]

Also, Spade asserts earlier that he's not quite what he seems when he says "Don't be too sure I'm as crooked as I'm supposed to be. That sort of reputation might be good business - bringing high-price jobs and making it easier to deal with the enemy."

On more than one occasion, Spade states that his main goal is to keep his own neck out of the noose by not becoming the fall guy for Archer's death.

If you've seen the film more than once, however, you know that Spade ultimately also wants to bring his partner's killer to justice. Making him seem indifferent to Archer's death is a way of keeping an unwitting audience guessing as to whether or not he will ultimately succumb to his feelings for Brigid. It's also why he's established as a man of loose morals when it comes to his partner's wife. The path he'll take is not clear.

Part of the film's fascination is the fact that Sam himself may not know what he's going to do until the last moment. His final "discussion" with Brigid is not with her but with himself. That's why he tells her not to talk.

The implication is that Spade doesn't like or care about Archer on a personal level (it's said outright in the book I think). Finding out who killed him is a professional matter, not a personal one.

Poor acting in that reaction. It was as emotional as reacting to a weather forecast

He's being cagey....nobody expects what ends he'll go to in order to get Miles' killer, even if it turns out to be the dame he loves.

"Why is spade so indifferent when hearing his partner is killed?" Because Huston and Bogart weren't quite sure what they were doing with Spade's character in this movie.

Because he doesn't show his feelings / cards to us as an audience. We are supposed to be guessing throughout this movie if he is a good guy or not, he rides that line, which is how he manages to gain the trust of the bad guys and reveal what they are up to.