Хотя врач и сказала, что царапина маленькая, я всё равно заклеил её пластырем.

Yes, врач is grammatically a masculine noun in its dictionary form, even when it refers to a woman.

In real usage, though, if the doctor is female, the past-tense verb is very often feminine:

• врач сказала = the doctor said, and the doctor is a woman
• врач сказал = the doctor said, and the doctor is a man

So here сказала tells you that the doctor was female.

This is common with profession words like врач, профессор, директор, etc. The noun itself may stay in the masculine form, but the verb can reflect the person’s real gender.

Хотя means although.

The extra и is a particle that strengthens the idea of concession. So:

• Хотя врач сказала... = Although the doctor said...
• Хотя врач и сказала... = Although the doctor did say...

In English, the second version can sound a bit like even though the doctor said... or although the doctor did say...

The и is not required, but it is very common in this pattern.

Because что introduces the content of what was said. It works like English that:

• сказала, что... = said that...

So:

• врач сказала, что царапина маленькая
  = the doctor said that the scratch was small

In Russian, что is very commonly used after verbs like:

• сказать = to say
• думать = to think
• знать = to know
• видеть = to see

Because царапина ends in -а, which is a very common ending for feminine nouns.

So it behaves like a feminine noun:

• царапина маленькая
• я заклеил её
• эта царапина

This also affects the pronoun and adjective forms used with it.

Because маленькая has to agree with царапина.

Царапина is:

• feminine
• singular
• nominative

So the adjective must also be:

• feminine
• singular
• nominative

That gives маленькая.

Compare:

• шрам маленький = the scar is small
• царапина маленькая = the scratch is small
• пятно маленькое = the spot is small

Here всё равно means anyway, all the same, or regardless.

So the idea is:

• The doctor said the scratch was small.
• Despite that, I still put a plaster on it.

That is why всё равно fits well after the concessive clause with хотя.

A very literal sense is something like it’s all the same, but in real English here the best translation is usually anyway or still.

Заклеить means to stick something over, to cover/seal with something adhesive.

With a scratch, that makes perfect sense: you put a plaster over it.

So:

• клеить = to glue / stick
• заклеить = to stick over, cover up with something sticky

Also, заклеил is perfective, so it shows a completed action:

• я заклеил её пластырем = I covered it with a plaster / I put a plaster on it

If you used an imperfective form, it would sound more like process, repetition, or background action, not a single completed act.

Because the sentence describes one completed action.

• заклеил = perfective, one completed act
• заклеивал = imperfective, process/repeated action/background

Here the speaker means:

• I went ahead and put a plaster on it

So the perfective past is the natural choice.

Её means her/it, and here it means it, referring back to царапина.

Since царапина is a feminine singular noun, the pronoun is feminine singular too.

In this sentence, её is the direct object of заклеил, so it means:

• заклеил её = covered it

For feminine singular nouns, the accusative pronoun form is её.

Because Russian often uses the instrumental case for the tool, means, or material used to do something.

Here:

• заклеить что? = to cover what?
• заклеить чем? = to cover with what?

So:

• заклеил её пластырем = covered it with a plaster

The noun пластырь becomes пластырем in the instrumental singular.

This is a very common pattern in Russian:

• писать ручкой = write with a pen
• резать ножом = cut with a knife
• накрыть одеялом = cover with a blanket
• заклеить пластырем = cover with a plaster

Because заклеил is masculine past tense.

In Russian past tense, verbs agree with the subject’s gender in the singular:

• я заклеил = I did it, spoken by a man
• я заклеила = I did it, spoken by a woman

So this sentence suggests that the speaker is male.

Yes. Russian word order is flexible, as long as the grammar stays clear.

This sentence could also be:

• Я всё равно заклеил её пластырем, хотя врач и сказала, что царапина маленькая.

That means the same thing. The difference is mainly emphasis:

• Starting with Хотя... foregrounds the contrast first.
• Starting with Я всё равно заклеил... foregrounds the speaker’s action first.

Both are natural.

Because the sentence contains subordinate clauses.

Structure:

• Хотя врач и сказала = concessive subordinate clause
• что царапина маленькая = content clause inside it
• я всё равно заклеил её пластырем = main clause

So the commas separate the clauses:

• after сказала because что царапина маленькая begins
• after маленькая because the хотя clause ends and the main clause begins

Russian punctuation is very clause-based, so commas appear often where English might also use them, especially with что, хотя, когда, если, and similar words.