После падения у меня болит колено.

Because после requires the genitive case.

The dictionary form is падение, but after после it changes to падения.

This is a very common pattern:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

So после падения literally means after the fall / after falling.

Падение is a noun meaning fall or falling.

It is related to the verbs:

• падать = to fall, to be falling, to fall repeatedly/ongoingly
• упасть = to fall down, to have a fall

So:

• Я упал(а) = I fell
• падение = a fall

Russian often uses a noun like падение after prepositions such as после, because it makes a compact phrase: после падения.

This is a very common Russian pattern for pain, illness, and physical states.

Russian often says:

• У меня болит голова = My head hurts
• У него болят зубы = His teeth hurt
• У нас болит спина = Our back hurts

Literally, у меня means something like at me or with me, but in natural English it corresponds to my in this kind of sentence.

So у меня болит колено is the normal way to say my knee hurts.

Моё колено болит is not impossible, but it sounds more marked or contrastive, as if you are emphasizing my knee specifically.

Because у requires the genitive case.

The pronoun я changes like this:

• nominative: я = I
• genitive: меня

So:

• у меня = at me / with me
• у тебя = at you / with you
• у него = at him / with him

This is just the normal grammar of the preposition у.

Because the pain is happening now.

The phrase после падения tells you that the fall happened earlier, but болит describes the current state:

• После падения = after the fall
• болит = hurts / is hurting

So the idea is:

After the fall, my knee hurts now.

If you wanted to say it hurt in the past, you could say:

• После падения у меня болело колено = After the fall, my knee was hurting / hurt

Because the subject is singular: колено = knee.

With болеть in this meaning, the verb agrees with the thing that hurts:

• болит колено = the knee hurts
• болит голова = the head hurts
• болят колени = the knees hurt
• болят зубы = the teeth hurt

So:

• singular body part → болит
• plural body parts → болят

Because колено is the grammatical subject of болит.

Russian structures this idea differently from English. In English, learners sometimes think in terms of I have knee pain or my knee hurts, but in Russian the body part itself is usually treated as the subject:

• колено болит = the knee hurts

Then у меня tells you whose knee it is.

So:

• у меня = whose situation it is
• колено = the thing that hurts
• болит = agrees with колено

No, Russian word order is fairly flexible.

После падения у меня болит колено is a very natural, neutral order.

But other orders are also possible, with different emphasis:

• У меня после падения болит колено
  Focuses a little more on me / my situation.

• Колено у меня болит после падения
  Can sound more conversational or emphasize the knee.

Russian word order often changes what is being highlighted, not the basic meaning.

Yes, absolutely.

That means:

After I fell, my knee hurts.

This version is more explicit because it uses a whole clause:

• после падения = after the fall / after falling
• после того, как я упал(а) = after I fell

Both are correct. The version with падения is shorter and more compact.

Also note:

• упал if the speaker is male
• упала if the speaker is female

Yes.

• колено = knee, neutral and standard
• коленка = knee, more colloquial and a bit more informal

So both are possible:

• У меня болит колено
• У меня болит коленка

The second one can sound a little more everyday or conversational, especially in speech.