С гипсом на ноге мне пришлось взять костыль, потому что без него я едва дохожу до кухни.

After the preposition с in the meaning with, Russian usually uses the instrumental case. That is why гипс becomes гипсом.

So:

• с гипсом = with a cast
• с гипсом на ноге = with a cast on my leg

This whole phrase describes the speaker’s condition.

Because на ноге describes location/state, not motion.

Compare:

• на ноге = on the leg
• на ногу = onto the leg

In this sentence, the cast is already on the leg, so Russian uses the prepositional case:

• нога → на ноге

Also, Russian often leaves out possessives like my when they are obvious from context, so на ноге naturally means on my leg here.

Мне пришлось means something like:

• I had to
• it turned out that I had to
• I was forced by circumstances to

The structure is impersonal in Russian. Literally, it is closer to to me, it became necessary.

That is why the person affected is in the dative case:

• я → мне

This pattern is very common:

• мне пришлось уйти = I had to leave
• ему пришлось ждать = he had to wait

Взять is perfective, so it focuses on a single completed action: the speaker ended up taking a crutch.

• взять = to take once, as a completed event
• брать = to take / be taking / take repeatedly

Here the meaning is: because of the situation, the speaker had to actually get a crutch. That is why взять fits better.

Not necessarily. Костыль can mean a crutch, and in real life someone may use one or two depending on the injury.

So:

• взять костыль = to take a crutch
• взять костыли = to take crutches

The singular here is completely natural if the speaker means one crutch.

Без него means without it, and него refers back to костыль.

The preposition без requires the genitive case, so:

• он → него

Here него does not mean him in the personal sense. It means it, referring to the masculine noun костыль.

So:

• без него = without it
• specifically: without the crutch

Доходить / дойти до means to reach, to make it as far as some place.

So:

• я дохожу до кухни = I make it to the kitchen
• я едва дохожу до кухни = I can barely make it to the kitchen

The verb доходить here highlights the idea of managing to reach a destination, not just walking in general.

Compare:

• я иду = I am walking / I go
• я дохожу до кухни = I make it as far as the kitchen

That extra idea of reaching the limit/destination is important in this sentence.

Because the sentence describes a current, repeated, or typical situation, not one future completed event.

• я едва дохожу до кухни = I can barely make it to the kitchen
  This sounds like an ongoing reality of the speaker’s condition.
• я едва дойду до кухни = I will barely make it to the kitchen
  This would point to one future attempt.

So the imperfective дохожу is used to describe what generally happens now.

The preposition до takes the genitive case and means up to / as far as / to in the sense of reaching a point.

So:

• кухня → кухни
• до кухни = to the kitchen / as far as the kitchen

This is a very common pattern:

• до дома = to the house / home
• до конца = to the end
• до остановки = to the bus stop

Едва means barely, hardly.

In this sentence it shows that the speaker can only just manage to reach the kitchen.

• я едва дохожу до кухни = I barely make it to the kitchen

It often adds a sense of difficulty, weakness, or very small degree.

Yes, the word order is natural.

Starting with С гипсом на ноге puts the speaker’s physical condition first, which helps explain everything that follows.

A rough information flow is:

1. With a cast on my leg...
2. I had to take a crutch...
3. because without it...
4. I can barely make it to the kitchen.

Russian word order is flexible, and the beginning of the sentence often sets the scene or highlights the most relevant circumstance.

It could, but the sentence as written is perfectly natural without it.

• С гипсом на ноге presents the situation: with a cast on my leg
• мне пришлось взять костыль then gives the consequence

If you wanted to stress the cause more explicitly, you might say something like:

• Из-за гипса на ноге мне пришлось взять костыль

That would mean Because of the cast on my leg, I had to take a crutch.

The original version sounds a bit more descriptive and less direct about cause, which is very natural Russian.