Learn Hardcore Malay: Saya berasa sedikit letih hari ini.

Berasa here means to feel (a certain way), both physically or emotionally.
You will very often hear Saya rasa sedikit letih hari ini instead; rasa in this kind of sentence is very common in everyday speech.
Grammatically, berasa is slightly more formal or standard, while rasa is more casual, but both are acceptable and understood as I feel.

Is berasa necessary? Can I just say Saya sedikit letih hari ini?

You can say Saya sedikit letih hari ini, and it is still understandable and grammatical.
However, adding berasa (or rasa) makes the sentence sound more natural because it clearly marks feel as a verb: I feel a bit tired today.
Without berasa, the sentence is more like I am a bit tired today, which is fine but slightly less typical in many contexts.

Why is sedikit placed before letih? Can I say letih sedikit instead?

In Malay, quantity words like sedikit (a little, a bit) usually come before adjectives and verbs, so sedikit letih is the most neutral order.
You can say letih sedikit; that tends to sound more like tired, but only a little and can give a slight emphasis on tired first, then limiting it.
Both are understood, but sedikit letih is more typical in a simple statement like this.

What is the difference between letih and penat? Could I say Saya berasa sedikit penat hari ini?

Yes, you can say Saya berasa sedikit penat hari ini and it means basically the same thing.
Letih and penat are very close in meaning: tired, weary, fatigued.
In many situations they are interchangeable; some speakers feel penat is more everyday and letih can sound a bit more formal or literary, but this varies by region and context.

Does hari ini show that the sentence is in the present tense? How does Malay show tense here?

Malay does not change the verb form for tense, so berasa itself does not show past, present, or future.
Time is usually shown by time expressions like hari ini (today), semalam (yesterday), esok (tomorrow), or by context.
So Saya berasa sedikit letih hari ini is understood as I feel a bit tired today (present) mainly because of hari ini.

Can I move hari ini to the front and say Hari ini saya berasa sedikit letih?

Yes, Hari ini saya berasa sedikit letih is completely correct and natural.
Time expressions such as hari ini can come at the beginning or at the end of the sentence.
Putting Hari ini first slightly emphasizes today: As for today, I feel a bit tired.

Can I drop Saya and just say Berasa sedikit letih hari ini?

You can hear Berasa sedikit letih hari ini in informal speech, especially when it is already clear from context that you are talking about yourself.
However, for clear, standard sentences (especially in writing or when learning), it is safer and more natural to include Saya.
So the full form Saya berasa sedikit letih hari ini is recommended for learners.

Is this sentence formal, informal, or neutral? How would a casual spoken version sound?

Saya berasa sedikit letih hari ini is neutral to slightly formal because of berasa and sedikit.
In casual spoken Malay, many people might say something like Saya rasa saya penat sikit hari ini or even Penat sikit hari ini.
The original sentence is perfectly fine in everyday conversation, especially if you usually speak in a more standard style.

Are there other words I could use instead of sedikit to mean a bit or a little?

Yes. Common alternatives include:

sikit – very frequent in speech: Saya rasa penat sikit hari ini.
agak – more like rather / somewhat: Saya agak letih hari ini.
kurang sihat (literally less healthy) used as a soft way of saying you feel a bit unwell: Saya rasa kurang sihat hari ini.
Sedikit is more neutral and slightly more formal than sikit, but they mean the same thing.