Eu escrevo o meu próprio nome no caderno.

In European Portuguese it’s standard to place the definite article (o/a) before possessive adjectives: o meu, a minha, o teu, etc. Omitting the article (​Escrevo meu próprio nome no caderno) sounds more like Brazilian usage or very informal speech. In Portugal you normally keep the article.

Portuguese is a “pro-drop” language, so the verb ending -o in escrevo already signals first person singular. You can safely omit Eu unless you want to emphasize who is doing the action or for clarity.

Próprio adds emphasis: it literally means own. By saying o meu próprio nome you stress that it’s your very own name (not someone else’s). If you drop próprio (Escrevo o meu nome…), people understand it’s your name, but you lose that extra emphasis.

No.

• Nome próprio (noun + adjective) is a fixed term meaning proper noun (e.g. Ana, Portugal).
• O meu próprio nome (possessive + próprio + noun) means my own name.
  Word order changes the meaning completely.

In Portuguese prepositions often merge with definite articles:

• em + o → no
• em + a → na
  So escrevo em o caderno becomes escrevo no caderno.

Yes.

• no caderno assumes context (the notebook in question is clear).
• no meu caderno explicitly means in my notebook.
  Use no meu caderno when you want to stress that it’s your notebook.

European Portuguese usually uses estar a + infinitive for ongoing actions. You’d say:
Estou a escrever o meu próprio nome no caderno.
The gerund form (estou escrevendo) is more typical of Brazilian Portuguese.