at have en klump i halsen

Yes, but only if it's the kind of happiness that makes you want to cry, like pride or being overwhelmed by kindness.

Always 'i halsen'. Using 'min' sounds like a direct translation from English and is not idiomatic in Danish.

It's neutral. You can use it with your friends or in a formal speech at a gala.

'Have' is the state of feeling it. 'Få' is the moment the feeling starts.

Yes, you can use adjectives like 'kæmpe' (huge) or 'lille' (small) to describe the intensity.

No, it's purely figurative, though it describes a real physical sensation.

Not really, but you might say 'Jeg blev helt ramt' (I was totally hit/affected).

Only if the meeting is very emotional (e.g., someone is retiring). Otherwise, it's too personal.

Extremely common in fiction to describe a character's emotional state.

Yes, though they might just say they are sad. They understand it when adults say it.

Go to the doctor! And use the words 'en hævelse' (a swelling) or 'en knude' (a node/lump) instead.

Yes, many Danish pop and folk songs use this to describe nostalgia.