Գետինը մտնել

No, it's not rude at all. It's a very common and expressive way to show that you are humble enough to feel shame.

It's a bit informal for a standard email, but if you've made a big mistake and have a good relationship with the person, it can show sincere regret.

'Ամաչել' is the simple verb 'to be ashamed'. 'Գետինը մտնել' is much stronger and more descriptive.

We entered the ground = Մենք գետինը մտանք (Menq getiny mtanq).

Yes, though the pronunciation might be 'Getinn antsnil' or 'Getinn mtnel' with a Western accent.

Not really. It's for a specific moment of embarrassment, not for a general personality trait like shyness.

No, you can say 'Նա գետինը մտավ' (He entered the ground) to describe someone else's embarrassment.

Young people might just say 'Գետինն եմ' (I'm the ground) as a very short, slangy way to say they are dying of shame.

No, it is strictly for negative social feelings like shame or embarrassment.

Then you would say 'Ընկա փոսը'. If you use 'Գետինը մտա', people will think you are talking about your feelings, not the hole!

Yes, it shows you respect social norms and understand when you've done something 'amot'.

Yes! It's often used self-deprecatingly to make people laugh after you've done something silly.