Läkaren sa att salvan skulle hjälpa om såret blev torrt.

Those endings are the Swedish definite article attached to the noun.

• en läkare = a doctor → läkaren = the doctor
• en salva = an ointment / a salve → salvan = the ointment
• ett sår = a wound → såret = the wound

Swedish usually puts the at the end of the noun instead of using a separate word like English does.

Here, om means if.

Swedish om can mean two different things:

• if: Om det regnar, stannar vi hemma = If it rains, we stay home
• about: en bok om Sverige = a book about Sweden

In your sentence, om såret blev torrt introduces a condition, so it means if the wound became/got dry.

Att here works like English that in reported speech.

• Läkaren sa att ... = The doctor said that ...

English often drops that, but Swedish commonly keeps att in sentences like this.

So:

• Läkaren sa att salvan skulle hjälpa
  means
• The doctor said that the ointment would help

In casual speech, att can sometimes be left out, but keeping it is very normal and clear.

Skulle hjälpa means would help.

This is used because the doctor was talking about a possible result, not stating that the ointment already helped.

Compare:

• hjälper = helps / will help
• hjälpte = helped
• skulle hjälpa = would help

Here skulle shows something expected or conditional from a past point of view:

• Läkaren sa att salvan skulle hjälpa ...
  = The doctor said that the ointment would help ...

So it is not describing a completed action. It is describing what the ointment was expected to do.

Here it means would, not should.

That is a very common point of confusion, because skulle can sometimes look like English should. But in this sentence:

• salvan skulle hjälpa = the ointment would help

It does not mean obligation or advice. It is about an expected result.

Because this sentence is built as a conditional/reporting structure.

• blir = becomes / gets
• blev = past form of bli, but in sentences like this it can also appear in a more hypothetical or backshifted conditional pattern

Swedish often uses:

• skulle in the main part
• a past tense form in the om clause

So:

• salvan skulle hjälpa om såret blev torrt

is a natural pattern meaning roughly:

• the ointment would help if the wound got/became dry

If you said:

• salvan hjälper om såret blir torrt

that would sound more like a direct, straightforward present/future statement:

• the ointment helps / will help if the wound gets dry

Because sår is a neuter noun: ett sår.

Predicative adjectives in Swedish agree with the noun:

• common gender singular: torr
• neuter singular: torrt
• plural: torra

So:

• ett sår → såret blev torrt

Compare:

• en hand blev torr = a hand became dry
• ett sår blev torrt = a wound became dry

The -t is there because såret is singular neuter.

Because these are subordinate clauses.

Swedish main clauses usually have the finite verb in second position, but subordinate clauses introduced by words like att and om usually have more straightforward word order:

• att + subject + verb
• om + subject + verb

So here:

• att salvan skulle hjälpa
• om såret blev torrt

This is normal subordinate-clause word order.

You can think of the whole sentence like this:

• Main clause: Läkaren sa
• Subordinate clause: att salvan skulle hjälpa
• Conditional clause inside it: om såret blev torrt

Yes.

• sa is the past tense of säga = to say
• blev is the past tense of bli = to become / get

Their present forms are:

• säger = says
• blir = becomes / gets

So the sentence uses past/reporting forms in a natural way:

• Läkaren sa ... = The doctor said ...
• såret blev torrt = the wound became/got dry

Not exactly. Blev torrt means became dry or got dry, not simply was dry.

• var torrt = was dry
• blev torrt = became / got dry

So bli focuses on a change of state.

That is why såret blev torrt suggests that the wound turned dry, not that it simply existed in a dry state already.