este miercuri dimineata

I am writing in my notebook early today (for that?) I am really bored. Also, I am very nervous about my test I will receive today. I hope I do well.

About the exam: did it already take place (a few days ago) and you are waiting to get the result?  Or the exam will take place today? 

In Romanian "a primi un examen"  makes no sense. You can say "a da un examen" (to participate to the exam)  or "a primi rezultatul [unui examen]" (to get the result [of an exam] ).

"a lua examenul" is also possible  -   "to pass/take the exam" (succeed) .

1. When do I know when to use "îmi?" I thought if I just used the "I" conjugation it is automatically know I am referring to me (I am writing).

I can understand why it is strange for you to see that the dative "îmi" needs to be used in a context where you are expecting to use posession "my notebook".

When we use the verb "a scrie"   and we want to say who is the beneficiary of the writing ... we have to use the dative îmi / îți /îi / ne / vă / le .

eu îmi scriu tema = i am writing the homework (the homework is for me!)

eu îți scriu tema = i am writing your homework (it is your homework, but i am helping you .  I write the homework instead of you,  even though  you are the one who should have written it. )

eu îi scriu o poezie = i am writing a poem for her ( i write the poem,  but it is dedicated to her  !)

There are also other verbs used with dative in a similar way

bunica îmi spune (îmi citește) o poveste = grandma is telling(reading) a story for me

To whom is grandma reading?  She is reading to me --> dative

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The sentence that we are analysing is somewhat tricky :  it is is not possible to translate "i am writing my notebook"  into Romanian in a word-by-word manner .

The "notebook" would have to be translated as "caiet" which means the entire physical booklet containing a certain number of empty pages. 

And you can't say "eu scriu un caiet"  because ou can't fill in the entire notebook at once !   You can write only on one page at a time.

Instead we say "eu scriu în caiet" = I write inside the notebook  (this probably sounds horribly in English ...)

But anyway, "caiet" can't be used to refer to this "notebook" feature of italki  ("caiet" can only be used for the physical notebook) . 

That's why Mihaela has changed the word to "notiță" (=note)  and reworded the entire sentence so that it sounds "natural".

2. Why examenului?

The preposition "din cauza" requires that the substantive has the genitive form (possession).

Translated literally in English it's something like "the reason why i am nervous is  the exam's fault " .  

The same happens for pronouns :

"din cauza mea"  -  my fault

"din cauza ta"  -  your fault

"din cauza lui"  -  his fault

etc..

("din cauza" is a preposition.    But we have also the noun "cauză" which is feminine : that's why "din cauza" requires the feminine posessive pronouns). 

Keep in mind that being a native English speaker is a disadavantage because in English the grammar is minimalistic when it comes to the cases of nouns/pronouns  (the plural is trivial ;  the genitive is " 's "  ;    the dative is " to + noun " ;  and that's all ). And you have no different word endings to worry about.

3. Again for "îl," the same question I have is for number 1. How do I know when to use it.

"că îl voi primi astăzi" -  is a sub-sentence.

The verb "a primi" requires that we specify "what exactly will be recieved".

So we use "îl"  - "că îl voi primi"   instead of repeating the noun "că voi primi examenul". 

Este miercuri dimineaţă

Scriu blocnotesul notiţa (mea)/ îmi scriu notiţa devreme astăzi pentru că sunt foarte plictisit. De asemenea, sunt foarte nervos (do you mean "nervous" from english, don't you? because you should have said "emoţionat". "nervos" means "mad" ) despre din cauza examenului meu că îl voi primi astăzi. (what do you want to say? that you will see your reasult today? or you'll take the exam today?) Sper să fac bine... (so you'll take the exam. you have to say " voi da examenul" )

Asta Săptămâna asta a fost foarte stressantă.

Succes la examen. :)