It's literally a core facet of dumbledores character that he's always genial and calm, even in the face of total disaster. It's part of what made him an intimidating wizard.
Even besides that though, why wouldn't he ask calmly? One of two things are true
-
Harry did put his name in the goblet and is lying about it, in which case going full on "DIDYOUPUTYOURNAMEINTHEGOBLETOFFIRE!?!?" is only likely to put Harry on the defensive and make him less likely to own up.
-
Harry didnt put his name in the goblet and is being setup by voldy or his buddies. In which case Harry is a victim and should be handled as such. At this point Dumbledore had very good reasons to think this was the more likely of the two.
Dont get me wrong, if you wanna talk about JKRs shitty writing, I'll happily join you, but this is actually an example of good writing on her part, given that the character in question is supposed to be the smartest wizard alive, it makes perfect sense that he'd react rationally instead of lashing out emotionally
You don't. Part of love is openness, you can't love a person while not being willing to share the fact that you love them. Pretty much by definition