Jeg skriver navnet mitt på skjemaet.

Both are grammatically correct. Norwegian has two patterns for possessives:

• Postposed (more neutral, most common): navnet mitt = my name
• Preposed (more formal/contrastive): mitt navn

So Jeg skriver navnet mitt sounds the most natural in everyday speech, while Jeg skriver mitt navn can sound a bit formal or emphatic.

Because navn is a neuter noun: et navn. The possessive must agree with the noun’s gender/number:

• masculine/common: min
• feminine: mi (in Bokmål many also use min)
• neuter: mitt
• plural: mine

Hence: navnet mitt.

Both can occur, but:

• på skjemaet is the default for writing on a form (think “on the page/sheet/surface”).
• i skjemaet is also used, especially when you mean “in the fields of the form,” often for digital forms.

Yes. Norwegian present tense covers both simple and progressive:

• Jeg skriver navnet mitt = I write / I am writing my name.

Use:

• Jeg signerer skjemaet.
• Jeg skriver under på skjemaet. (Also common: underskrive/undertegne.)

Default word order is Subject–Verb–Object–(Adverbials): Jeg skriver navnet mitt på skjemaet. For emphasis you can front the adverbial, but remember Norwegian V2 (the finite verb must be in second position):

• På skjemaet skriver jeg navnet mitt. Putting it as Jeg skriver på skjemaet navnet mitt is possible but sounds marked/unusual.

mitt is first‑person possessive (my).
sitt is the third‑person reflexive possessive (his/her/their own, neuter). For example:

• Han skriver navnet sitt = He writes his own name.
• Han skriver navnet hans = He writes his name (someone else’s).

Use fylle ut with the form itself:

• Jeg fyller ut skjemaet. = I’m filling out the form. To mention your name as part of that, you’d say:
• Jeg fyller ut skjemaet med navnet mitt og adressen min. But don’t say fylle ut navnet mitt; you skriver a name.