Я наклеила пластырь на пятку и смогла дойти до дома пешком.

Because the speaker is female. In the Russian past tense, verbs agree with the subject in gender and number.

• наклеила = feminine singular past
• смогла = feminine singular past

If a man said the same sentence, it would be:

Я наклеил пластырь на пятку и смог дойти до дома пешком.

It actually is in the accusative case, but for many inanimate masculine singular nouns, the accusative looks exactly like the nominative.

So here:

• nominative: пластырь
• accusative: пластырь

That is why you see no visible ending change.

Because after a verb of putting or attaching something onto a surface, Russian usually uses на + accusative.

Here the idea is stick/apply something onto the heel, so:

• пятка = nominative
• пятку = accusative singular

Compare:

• книга на столе = the book is on the table
• положить книгу на стол = to put the book onto the table

In your sentence, the plaster is being placed onto the heel, so на пятку is correct.

Because наклеила is perfective, so it presents the action as completed: she put the plaster on.

• клеила would sound like she was in the process of sticking it on, or was doing that action in a more ongoing way
• наклеила means the action was successfully completed

That fits the sentence, because the next part gives the result: after doing that, she managed to walk home.

Пластырь is a broad word. Depending on context, it can mean:

• a sticking plaster
• an adhesive bandage
• a Band-Aid
• sometimes medical adhesive tape

In this sentence, because of на пятку, it most likely means a plaster/bandage put on the heel, probably because of a blister or sore spot.

Because смогла means managed to or succeeded in doing something in this specific situation.

• могла often means could / was able to in a more general or potential sense
• смогла focuses on the fact that she actually did it successfully

So смогла дойти strongly suggests a successful result: she really did make it home.

After смочь, Russian uses an infinitive: смогла дойти = managed to walk/reach.

The verb дойти is important here because it means to get as far as / to reach on foot. It is not just the process of walking; it includes reaching the destination.

That is why дойти works especially well here: the speaker had trouble with her heel, but still managed to make it home.

Because идти just means to go / to be walking and does not by itself show completion.

• идти = to walk, to be going
• дойти = to get there on foot, to make it as far as

Since the sentence is about a completed achievement, дойти is the better choice.

After до, Russian uses the genitive case.

So:

• дом = nominative
• дома = genitive singular

The pattern is дойти до + genitive = to reach as far as something.

So дойти до дома means to make it to the house / home.

It looks the same, but here it is functioning differently.

In this sentence, дома is the genitive singular form of дом after the preposition до.

So:

• Я дома = I am at home
• до дома = to the house / as far as home

The forms are identical in spelling, but the grammar is different.

Often yes. Дойти домой is also natural.

But дойти до дома follows a very common pattern with дойти до + genitive and can sound a bit more explicit about reaching the endpoint.

So both are possible, but дойти до дома is very standard and natural here.

Because пешком is a fixed adverbial form meaning on foot.

Russian does not say something like на ногах for this meaning in normal neutral speech. It simply uses:

• идти пешком = go on foot
• дойти пешком = reach on foot

Historically this form is related to the instrumental case, but for a learner it is easiest to treat пешком as a set word meaning on foot.

Because the subject is still the same, and Russian often leaves out repeated pronouns when they are obvious.

So:

Я наклеила пластырь на пятку и смогла дойти до дома пешком.

is perfectly natural.

You could repeat я, but it would usually add emphasis or sound less smooth:

Я наклеила пластырь на пятку, и я смогла дойти до дома пешком.

Normally, the version without the second я is better.