أنا أريد أن أذهب إلى المدرسة اليوم.

Not usually. أنا (I) is often dropped because the verb already shows the subject:

• أريد أن أذهب إلى المدرسة اليوم. = (I) want to go to school today.
  You keep أنا when you want emphasis or contrast (e.g., أنا أريد... = I want..., not someone else).

Arabic (including Modern Standard) doesn’t use an infinitive the way English does. Instead, it uses:

• أن + فعل مضارع (أن + present verb)
  So أن أذهب is the standard way to express to go in this structure.

Yes—formally in MSA, أن puts the following present verb into the subjunctive (المنصوب).
With full vowel marks:

• أُرِيدُ أَنْ أَذْهَبَ (notice أذهبَ ends with -a / fatḥa)
  In everyday unvowelled writing, you’ll still usually see أذهب without showing the ending.

A common fully vowelled version is:

• أَنَا أُرِيدُ أَنْ أَذْهَبَ إِلَى الْمَدْرَسَةِ الْيَوْمَ.
  Case endings can vary slightly depending on analysis and pause, but this is a solid “textbook MSA” reading.

Yes. That uses a verbal noun (مصدر):

• أريد الذهاب إلى المدرسة اليوم = I want going / I want to go to school today
  Both are correct; أريد أن أذهب... is often more direct and “verbal,” while أريد الذهاب... can sound a bit more nominal/formal.

Because إلى means to/toward a destination:

• أذهب إلى المدرسة = I go to the school.
  لِـ usually means for/to (for someone) or indicates purpose/possession in other contexts, and it doesn’t typically mark a physical destination with ذهب the same way إلى does.

اليوم (today) is flexible. Common options:

• ... إلى المدرسة اليوم (very natural)
• اليوم أريد أن أذهب إلى المدرسة (emphasizes “today”)
  Moving it changes emphasis, not basic meaning.

Two common ways:

• Present-time negation: لا أريد أن أذهب إلى المدرسة اليوم. = I don’t want to go to school today.
• More emphatic/general: لستُ أريد أن أذهب... is less common in this exact pattern; لا أريد... is the go-to choice.