Se miaj manoj estas malpuraj, mi lavas ilin denove.

Because the possessive adjective mia has to agree with the noun it describes.

• mano = hand
• manoj = hands

Since manoj is plural, mia becomes miaj.

So:

• mia mano = my hand
• miaj manoj = my hands

Esperanto adjectives and possessives match the noun in number, and sometimes also in case.

The ending -j marks the plural in Esperanto.

• mano = hand
• manoj = hands

Since the sentence talks about both hands, the plural is used.

Because malpuraj describes manoj, and adjectives must agree with the noun they modify.

• manoj is plural
• so the adjective must also be plural: malpuraj

Compare:

• malpura mano = a dirty hand
• malpuraj manoj = dirty hands

mal- is a very common Esperanto prefix meaning the opposite of the base word.

• pura = clean
• malpura = dirty

So malpuraj literally means unclean or dirty.

This prefix is very productive in Esperanto:

• bona = good → malbona = bad
• granda = big → malgranda = small

estas means are / is, so miaj manoj estas malpuraj means my hands are dirty.

This describes a state, not a change.

• estas malpuraj = are dirty
• iĝas malpuraj = become dirty / are getting dirty

So in this sentence, the speaker is saying that the hands are already dirty.

Because ilin is the direct object form of ili.

• ili = they
• ilin = them

In mi lavas ilin, ilin is the thing being washed, so it needs the -n ending for the direct object.

Compare:

• Ili estas malpuraj. = They are dirty.
• Mi lavas ilin. = I wash them.

It refers back to miaj manoj.

So:

• miaj manoj = my hands
• ilin = them

Esperanto often uses a pronoun like this instead of repeating the noun:

• Se miaj manoj estas malpuraj, mi lavas miajn manojn denove.
• Se miaj manoj estas malpuraj, mi lavas ilin denove.

Both are possible. Using ilin is a little more natural because it avoids repetition.

Because in the first part, miaj manoj is the subject, not the direct object.

• miaj manoj estas malpuraj = my hands are dirty
  Here, miaj manoj is the subject, so there is no -n.

In the second part, if you repeated the noun, it would be:

• mi lavas miajn manojn = I wash my hands

Here miajn manojn is the direct object, so both words take the right endings:

• miajn = plural + accusative
• manojn = plural + accusative

Esperanto uses the present tense -as for present-time actions, habitual actions, and many general statements.

• mi lavas = I wash / I am washing

In a sentence with se (if), Esperanto often keeps the present tense in both parts when English does too:

• Se pluvas, mi restas hejme. = If it rains, I stay home.
• Se miaj manoj estas malpuraj, mi lavas ilin denove. = If my hands are dirty, I wash them again.

So lavas is completely normal here.

Denove means again or anew.

So mi lavas ilin denove means I wash them again.

It suggests that the speaker has already washed them before, or is repeating the action.

Yes. Esperanto word order is fairly flexible, as long as the meaning stays clear.

These are all possible:

• Mi lavas ilin denove.
• Mi denove lavas ilin.
• Denove mi lavas ilin.

The most neutral version here is probably mi lavas ilin denove.

Changing the position can slightly shift emphasis:

• Mi denove lavas ilin = I again wash them
• Denove mi lavas ilin = Again, I wash them

Se means if.

It introduces a condition:

• Se miaj manoj estas malpuraj = If my hands are dirty

Then the main clause says what happens in that case:

• mi lavas ilin denove = I wash them again

So the sentence structure is:

• Se
  • condition
• result/action

This works very much like English if.

A comma after the se clause is standard and helpful:

• Se miaj manoj estas malpuraj, mi lavas ilin denove.

Esperanto commonly uses a comma between a subordinate clause and the main clause, especially when the subordinate clause comes first. So the comma here is the normal choice.

The meaning is basically the same in this sentence.

• mi lavas ilin = I wash them
• mi lavas miajn manojn = I wash my hands

The version with ilin sounds smoother because it avoids repeating hands. The full noun phrase is more explicit, but repetition is usually unnecessary once the reference is clear.

Esperanto has no separate word for the. It uses la when needed, but possessives like mia, via, lia, and so on already make the noun definite enough in most cases.

So:

• miaj manoj = my hands

You do not say la miaj manoj in normal Esperanto.

Yes. Esperanto present tense does not force a distinction between:

• I wash them again
• I am washing them again

Both can be expressed by mi lavas ilin denove. Context tells you which is meant.

In this sentence, the general or habitual meaning is the most natural: If my hands are dirty, I wash them again.