Мне стало неловко, когда я понял, что написал имя получателя с ошибкой.

Because Russian often expresses feelings and states with an impersonal construction:

• мне = to me
• стало неловко = it became awkward / I felt awkward

So Мне стало неловко literally means something like To me it became awkward, but natural English is I felt awkward/embarrassed.

If you said я стал неловким, that would mean I became awkward/clumsy as a personal characteristic, which is different.

Стало is the past tense of стать and here it means became.

So:

• Мне было неловко = I felt awkward / I was embarrassed
• Мне стало неловко = I started to feel awkward / I became embarrassed

The sentence uses стало because it focuses on the moment when the feeling appeared.

Here неловко is not acting like a regular adjective. It is a predicative state word often used in impersonal sentences.

Compare:

• неловкий человек = an awkward person
• мне неловко = I feel awkward / embarrassed

So in this sentence, неловко describes a state, not a noun.

In the Russian past tense, singular verb forms show gender.

• я понял
• я написал

These forms are masculine, so the speaker is understood to be male.

If the speaker were female, the sentence would be:

• Мне стало неловко, когда я поняла, что написала имя получателя с ошибкой.

Both verbs describe completed events:

• понял = realized / understood at a certain moment
• написал = wrote / finished writing

That fits the meaning well:

1. the person completed writing the name,
2. then realized there was a mistake,
3. then felt embarrassed.

If you used imperfective forms, the meaning would change:

• понимал = was understanding / used to understand
• писал = was writing / used to write

Those would not fit this situation as naturally.

Что introduces the content of what was realized. It works like English that.

• я понял, что... = I realized that...

So the clause что написал имя получателя с ошибкой explains exactly what the speaker realized.

Получателя is genitive singular.

Russian often uses the genitive to show a relationship like of or English possessive 's:

• имя получателя = the recipient's name
• literally the name of the recipient

You can think of it as answering the question:

• имя кого? = whose name?
• получателя

Because the expression с ошибкой literally means with an error / with a mistake.

• написать имя с ошибкой = to write the name with a mistake
• natural English: to misspell the name or to write the name incorrectly

The noun ошибка comes after с, and in this meaning с takes the instrumental case, so:

• ошибка → ошибкой

A useful contrast:

• с ошибкой = there is an error in what was written
• по ошибке = by mistake, accidentally

Those are related, but not the same.

Yes. Russian word order is fairly flexible.

For example, you could also say:

• Когда я понял, что написал имя получателя с ошибкой, мне стало неловко.

That means the same thing.

The difference is mainly focus:

• Мне стало неловко... puts the emotional reaction first.
• Когда я понял... puts the situation first, then the reaction.

Both are natural.

Not quite.

• неловко = awkward, embarrassed, uncomfortable
• стыдно = ashamed

Неловко is usually milder and often fits social embarrassment.
Стыдно is stronger and can suggest real shame.

So in this sentence, неловко means the speaker felt awkward or embarrassed after noticing the mistake, not necessarily deeply ashamed.