Mit dem Daumen schließe ich den Gürtel, obwohl meine rechte Hand noch weh tut.

Because mit always takes the dative case, and Daumen is masculine singular.

• der Daumen = nominative
• den Daumen = accusative
• dem Daumen = dative

So after mit, you need:

• mit dem Daumen = with the thumb

If it were plural, it would be mit den Daumen, but here it is singular.

German often puts a phrase at the beginning of the sentence to give it emphasis or set the scene.

The normal basic order would be:

• Ich schließe den Gürtel mit dem Daumen.

But German can also front the prepositional phrase:

• Mit dem Daumen schließe ich den Gürtel.

This puts a little more focus on how the action is being done: with the thumb.

This is because German main clauses follow the verb-second rule.

The finite verb must be in the second position. If something other than the subject comes first, the verb still stays second, and the subject comes after it.

So:

• Ich schließe den Gürtel.
  subject first, verb second

But if you move Mit dem Daumen to the front:

• Mit dem Daumen schließe ich den Gürtel.

Now the first position is taken by Mit dem Daumen, so schließe must come next.

Because der Gürtel is the direct object of schließen, so it takes the accusative case.

• der Gürtel = nominative
• den Gürtel = accusative

Here, the speaker is fastening/closing the belt, so the belt is what the action is being done to.

Yes, but in this context it is better understood as to fasten, to close, or to do up.

For example:

• die Tür schließen = to close the door
• den Reißverschluss schließen = to close the zipper
• den Gürtel schließen = to fasten the belt

So schließen can be used for different things that are being closed or fastened.

Obwohl means although and introduces a subordinate clause.

In German subordinate clauses, the finite verb goes to the end.

So:

• ... obwohl meine rechte Hand noch weh tut.

Compare with a main clause:

• Meine rechte Hand tut noch weh.

After obwohl, the verb is pushed to the end:

• ..., obwohl meine rechte Hand noch weh tut.

The expression is wehtun, meaning to hurt.

In a normal main clause, it appears separated:

• Meine Hand tut weh. = My hand hurts.

In a subordinate clause, the conjugated verb goes to the end, so you get:

• ..., obwohl meine Hand weh tut.

So weh tun behaves like a separable expression in this sentence pattern.

The dictionary form is usually given as wehtun.

But in actual sentences, the finite forms are often split:

• Die Hand tut weh.
• Mein Arm tat weh.

With infinitives or participles, you often see the one-word form:

• Es kann wehtun.
• Das hat wehgetan.

So learners should recognize both:

• dictionary form: wehtun
• common finite form in sentences: tut weh

Because Hand is feminine, and here it is the subject of the subordinate clause, so it is in the nominative.

The noun phrase is:

• meine = my
• rechte = right
• Hand = hand

Since Hand is feminine nominative singular, the correct form is:

• meine rechte Hand

Not:

• meinen rechten Hand
  which would be wrong here

German often uses the definite article with body parts when it is already clear whose body part is meant.

So:

• mit dem Daumen = with the thumb

is very natural, even though English often prefers my thumb.

But in other places, German can also use a possessive when it helps clarify things:

• meine rechte Hand = my right hand

Here, meine helps specify exactly which hand is meant.

So both patterns are normal:

• der/dem + body part when ownership is obvious
• mein/dein/etc. + body part when the speaker wants to specify

Here noch means still.

So:

• meine rechte Hand noch weh tut
  = my right hand still hurts

It shows that the pain is continuing.

Compare:

• Meine Hand tut weh. = My hand hurts.
• Meine Hand tut noch weh. = My hand still hurts.

Because German normally puts a comma before a subordinate clause, and obwohl introduces one.

So the comma separates:

• main clause: Mit dem Daumen schließe ich den Gürtel
• subordinate clause: obwohl meine rechte Hand noch weh tut

This is standard German punctuation.

Both verbs are in the present tense:

• schließe = I fasten / I am fastening
• tut weh = hurts / is hurting

German uses the present tense very often for actions happening right now, where English might use either:

• I fasten the belt
• I am fastening the belt

So the German present tense can cover both a simple present and a present progressive meaning, depending on context.

Yes, absolutely.

That version is also correct:

• Ich schließe den Gürtel mit dem Daumen, obwohl meine rechte Hand noch weh tut.

The difference is mostly focus:

• Mit dem Daumen schließe ich den Gürtel...
  puts more emphasis on with the thumb

• Ich schließe den Gürtel mit dem Daumen...
  is a more neutral word order

Both are grammatical.