En tombant, je me suis fait une petite coupure au coude.

It is the gérondif form: en + present participle (tombant from tomber).

In this sentence, it means something like:

• while falling
• as I fell
• when I fell

It gives the circumstance of the main action: the cut happened during the fall.

This is a very common French pattern:

se faire + noun

In everyday French, it often means to get / to end up with / to sustain something, especially an injury.

So:

• se faire une coupure = to get a cut
• se faire mal = to hurt oneself
• se faire une entorse = to sprain something / get a sprain

Literally, je me suis fait une petite coupure is close to I made myself a small cut, but natural English is I got a small cut.

Because se faire is a reflexive/pronominal verb in this sentence, and reflexive verbs use être in the passé composé.

So:

• je me suis fait
• not je m’ai fait

This is the same pattern as:

• je me suis levé
• elle s’est blessée
• nous nous sommes trompés

Because in this sentence, fait does not agree with the subject.

The reason is that une petite coupure is the direct object, and it comes after the past participle. The reflexive pronoun me is not the direct object here.

So both a man and a woman would say:

• Je me suis fait une petite coupure au coude.

This is similar to:

• Elle s’est lavé les mains.
• Elle s’est cassé la jambe.

No extra -e is added to fait here.

Yes, you could, but the nuance is a little different.

• Je me suis fait une petite coupure au coude focuses on the resulting injury: I got a small cut on my elbow.
• Je me suis coupé le coude focuses more directly on the action: I cut my elbow / I cut myself on the elbow.

Both are natural, but se faire une coupure is especially common when talking about ending up with a cut.

Usually yes: une petite coupure means a small cut or a minor cut.

But French also often uses petit/petite in a slightly softening, everyday way. It can make something sound less serious or more casual.

For example:

• un petit rhume = a slight cold
• une petite douleur = a little pain
• une petite coupure = a minor cut

So here it probably means the injury was not serious.

au is the contraction of à + le:

• à le coude → au coude

So literally it is at the elbow / on the elbow, but natural English is on my elbow or to my elbow, depending on context.

French very often uses à with injuries or physical problems:

• une douleur au dos = back pain
• une blessure à la main = an injury to the hand
• une coupure au doigt = a cut on the finger

French usually uses the definite article with body parts when the owner is already clear.

Here, the owner is already shown by me in je me suis fait. So French says:

• au coude
• not usually à mon coude

This is a very common pattern:

• Je me suis lavé les mains.
• Il s’est cassé la jambe.
• Elle s’est touché le visage.

English prefers my, but French usually prefers the in this structure.

Yes, very natural.

French often starts with a short introductory phrase that explains the context:

• En courant, ... = while running...
• En sortant, ... = while going out...
• En tombant, ... = while falling...

It is a compact, natural way to say what was happening when the main event occurred. The comma after En tombant is also normal because it is an introductory phrase.