Meine rechte Hand tut weh.

Because Hand is a feminine noun in German: die Hand.

The possessive mein- changes its ending to match the noun’s gender, number, and case. In the nominative singular feminine, it becomes meine.

• mein Arm = my arm
• meine Hand = my hand

So meine rechte Hand is the correct form.

Because rechte is the adjective form that matches the noun phrase meine Hand.

In meine rechte Hand:

• meine already shows feminine nominative singular
• the adjective recht- takes the ending -e

So:

• die rechte Hand
• meine rechte Hand

By contrast:

• rechter would be used with some masculine forms
• rechts is usually an adverb, meaning on the right or to the right

Here, you need the adjective form rechte because it directly describes Hand.

In German, all nouns are capitalized.

So Hand gets a capital letter because it is a noun, just like:

• der Arm
• das Bein
• die Schulter

This is a standard rule in German spelling.

Wehtun is a German verb meaning to hurt.

It is often split in a sentence because it is a separable verb:

• dictionary form: wehtun
• in a normal main clause: tut ... weh

So:

• Meine rechte Hand tut weh.
• literally: My right hand does pain / hurts

The important thing is that tun is the conjugated part, and weh goes to the end of the clause.

Because the subject is Meine rechte Hand, which is third-person singular.

The verb is wehtun, and in the present tense:

• ich tue weh
• du tust weh
• er/sie/es tut weh
• wir tun weh
• ihr tut weh
• sie/Sie tun weh

Since Hand is singular, you use tut:

• Meine rechte Hand tut weh.

If the subject were plural, you would use tun:

• Meine Hände tun weh.

German often uses a possessive with body parts where English also commonly does:

• meine Hand = my hand
• mein Kopf = my head

So Meine rechte Hand tut weh is completely natural.

German can also sometimes use a definite article with body parts in other structures, especially when a dative pronoun appears:

• Mir tut die rechte Hand weh.

That also means My right hand hurts, but the structure is a little different:

• mir = to me
• die rechte Hand = the right hand

Both are natural, but Meine rechte Hand tut weh is straightforward and easy for learners.

Yes. That is a very common alternative.

Compare:

• Meine rechte Hand tut weh.
• Mir tut die rechte Hand weh.

Both mean the same thing in normal English: My right hand hurts.

The difference is grammatical:

• Meine rechte Hand tut weh.
  The subject is meine rechte Hand.

• Mir tut die rechte Hand weh.
  The subject is die rechte Hand, and mir is in the dative, meaning something like to me.

German often uses this dative structure for physical sensations.

Because wehtun is a separable verb.

In a main clause, the conjugated verb goes to the second position, and the separable prefix or particle goes to the end:

• Es tut weh.
• Meine rechte Hand tut weh.

In an infinitive, the parts stay together:

• Meine Hand kann wehtun.

So weh goes at the end here because that is how separable verbs work in German main clauses.

No. Another common verb is schmerzen:

• Meine rechte Hand schmerzt.

That also means My right hand hurts, but it sounds a bit more formal or clinical than tut weh.

In everyday spoken German, tut weh is extremely common and natural.

Yes. This pattern is very common.

Singular:

• Mein Rücken tut weh. = My back hurts.
• Mein Kopf tut weh. = My head hurts.

Plural:

• Meine Füße tun weh. = My feet hurt.
• Meine Hände tun weh. = My hands hurt.

Notice that the verb changes with singular vs. plural:

• singular: tut weh
• plural: tun weh

In a subordinate clause, the full verb wehtun stays together at the end.

For example:

• Ich glaube, dass meine rechte Hand wehtut.

That is different from the main clause:

• Meine rechte Hand tut weh.

So:

• main clause: tut ... weh
• subordinate clause: wehtut

This is a useful pattern to learn for separable verbs in general.