Because the verb is devoir, and dois is the je form in the present tense.
Present tense of devoir:
So with je, you must say je dois.
Because nom is a masculine singular noun, so the correct possessive adjective is mon.
French possessive adjectives must agree with the thing being possessed, not with the owner.
So:
Even if the speaker is female, she would still say mon nom, because nom is masculine.
Because sur is the normal way to say on in this context.
So:
Using dans le formulaire would usually sound less natural here. Dans means in/inside, and although English sometimes says in the form, French commonly uses sur for writing something on a document.
Le formulaire usually refers to a specific form that is already known from the situation.
For example, if someone hands you a form and says what you need to do, French often uses le:
If you were talking about any form in a general sense, un formulaire could be possible:
So le here suggests the form in question.
Yes, it is the most neutral and standard word order.
The structure is:
Je + dois + écrire + mon nom + sur le formulaire
That is:
French word order is often less flexible than English, so this is a very natural basic sentence.
A rough English-style guide would be:
zhuh dwah ay-kreer mohn nohn syur luh for-myoo-lair
A few useful details:
This kind of guide is only approximate, but it can help you get started.
It depends on context.
In everyday language, nom can simply mean name in general. But on official forms, French often distinguishes:
So in a real administrative context, mon nom might sometimes be understood specifically as a last name, depending on the form.
They are similar, but not identical.
The second structure is more advanced because it uses que plus the subjunctive: j’écrive. For learners, je dois + infinitive is usually the simpler pattern to start with.