Το δεξί μου χέρι πονάει σήμερα.

Because χέρι is a neuter singular noun, so the adjective has to match it.

• ο δεξιός = masculine
• η δεξιά = feminine
• το δεξί = neuter

So:

• το δεξί χέρι = the right hand

This is normal adjective agreement in Greek: the article, adjective, and noun all match in gender, number, and case.

In Greek, it is very common to use the definite article together with a possessive clitic like μου.

So Greek normally says:

• το χέρι μου = literally the hand my, meaning my hand

This is especially common with:

• body parts
• family members
• everyday personal nouns

So Το δεξί μου χέρι is the natural Greek way to say my right hand.

μου is an unstressed clitic form, so it does not behave like a full independent word such as English my.

In Greek noun phrases, μου usually comes after the noun or after an adjective within the phrase:

• το χέρι μου = my hand
• το δεξί μου χέρι = my right hand

You would not say:

• το μου δεξί χέρι

That sounds ungrammatical in standard Greek.

Here χέρι is in the nominative singular, because it is the subject of the verb πονάει.

The hand is the thing that hurts:

• Το δεξί μου χέρι πονάει.
• My right hand hurts.

A small extra point: χέρι has the same form in the nominative and accusative singular, so the form itself does not change here, but its role in the sentence is nominative.

πονάει is the 3rd person singular present form of the verb πονάω / πονώ, meaning to hurt or to be in pain.

So:

• πονάω = I hurt / I am hurting
• πονάς = you hurt
• πονάει = he/she/it hurts

In this sentence, the subject is το δεξί μου χέρι, which is singular, so πονάει is the correct form.

Yes. In modern Greek, both πονάει and πονά are common 3rd person singular forms in everyday speech.

So these both work:

• Το δεξί μου χέρι πονάει σήμερα.
• Το δεξί μου χέρι πονά σήμερα.

They mean the same thing.
πονάει may sound a little fuller or slightly more explicit, but both are perfectly normal.

Yes, Greek word order is fairly flexible, and σήμερα can move.

All of these are natural:

• Το δεξί μου χέρι πονάει σήμερα.
• Σήμερα πονάει το δεξί μου χέρι.
• Πονάει σήμερα το δεξί μου χέρι.

The meaning stays basically the same, but the emphasis can shift a little.

Putting σήμερα at the end is a very normal, neutral way to add the time expression.

Yes, you can, and it is very common.

• Το δεξί μου χέρι πονάει. = My right hand hurts.
• Με πονάει το δεξί μου χέρι. = literally My right hand hurts me, but in natural English still My right hand hurts.

The version with με adds the person who experiences the pain more explicitly. Greek often uses both patterns.

So both are correct and natural; the sentence you were given is simply a straightforward way to say it.