Мне всё равно, какой салон выбрать, лишь бы он был рядом с домом.

Because всё равно is an impersonal expression, and the person affected by it goes in the dative case.

So:

• мне всё равно = it’s all the same to me / I don’t care
• literally, something like to me, everything is equal

That is why Russian uses мне, not я.

Compare:

• Мне холодно = I’m cold
• Мне интересно = I’m interested
• Мне всё равно = I don’t care

In all of these, Russian expresses the idea as to me it is..., not with a nominative subject like English.

Here всё равно means I don’t mind / it makes no difference to me / I don’t care.

In this sentence, it means the speaker has no preference about which salon to choose.

A few close English equivalents are:

• I don’t care which salon to choose
• It doesn’t matter to me which salon to choose
• I’m fine with any salon

A few similar Russian expressions are:

• мне не важно = it’s not important to me
• мне без разницы = it makes no difference to me (more colloquial)
• мне всё равно = neutral and very common

Какой means which / what kind of.

In какой салон выбрать, it introduces an indirect question:

• какой салон выбрать = which salon to choose

So the whole first part means:

• Мне всё равно, какой салон выбрать = I don’t care which salon to choose

Grammatically, какой agrees with салон in gender, number, and case.

Since салон is masculine singular, какой is also masculine singular.

Good question. Салон is indeed the direct object of выбрать, so it is in the accusative case.

But for masculine inanimate nouns, the accusative singular looks exactly like the nominative singular.

So:

• nominative: какой салон
• accusative: какой салон

They are the same in form.

If the noun were feminine, you would see the difference more clearly. For example:

• какую машину выбрать = which car to choose

Here какую clearly shows the accusative.

Because Russian often uses an infinitive after expressions like мне всё равно, я не знаю, непонятно, etc. when talking about what to do.

So:

• какой салон выбрать = which salon to choose
• куда пойти = where to go
• что купить = what to buy

This is very natural in Russian.

You could think of the structure as:

• Мне всё равно, [какой салон выбрать].

That whole clause functions like which salon to choose in English.

Лишь бы means as long as, provided that, or sometimes if only depending on context.

In this sentence:

• лишь бы он был рядом с домом = as long as it is near the house

It introduces the only condition that matters to the speaker.

So the logic is:

• I don’t care which salon to choose
• as long as it’s close to home

This expression is very common in spoken and written Russian.

Examples:

• Ешь что хочешь, лишь бы был сыт. = Eat whatever you want, as long as you’re full.
• Мне всё равно, лишь бы не было дорого. = I don’t care, as long as it isn’t expensive.

After лишь бы, Russian commonly uses бы with a past-tense form to express a desired condition, not actual past time.

So:

• лишь бы он был... does not mean as long as he/it was... in the past
• it means as long as he/it be / is / would be...

This is a standard Russian pattern:

• лишь бы было тихо = as long as it’s quiet
• лишь бы пришёл = as long as he comes
• лишь бы всё получилось = as long as everything works out

So был here is part of a conditional/subjunctive-like construction with бы, not a simple past tense.

Он refers to салон.

Since салон is a masculine noun, the pronoun used to refer back to it is он.

So:

• салон = masculine
• он = it / he depending on context
• here it means it

Russian does not have a separate pronoun for it, so он, она, or оно are used depending on the gender of the noun.

Because the expression рядом с requires the instrumental case.

So:

• рядом с домом = next to / near the house
• с домом is instrumental because of рядом с

More examples:

• рядом с метро = near the metro
• рядом с магазином = near the store
• рядом с работой = near work

This is just a fixed pattern you need to learn:

• рядом с + instrumental

It can suggest near the house/home, not necessarily literally touching the house.

In real usage, рядом с домом often means close to where I live or near my house/home.

So in this sentence, the natural meaning is probably:

• close to home
• near my house

not necessarily immediately next door.

Russian рядом can be a little broader in everyday speech than strict English right next to.

Not necessarily. Салон can mean different kinds of salon/showroom/businesses, depending on context.

Common possibilities include:

• beauty salon
• hair salon
• car showroom/dealership
• furniture showroom
• other service businesses called салон

So without more context, салон is somewhat broad. The grammar of the sentence does not tell us exactly which type it is.

Yes, Russian word order is flexible, though some orders sound more natural than others.

The original:

• Мне всё равно, какой салон выбрать, лишь бы он был рядом с домом.

is very natural and neutral.

Possible variations include:

• Мне всё равно, какой выбрать салон, лишь бы он был рядом с домом.
• Какой салон выбрать, мне всё равно, лишь бы он был рядом с домом.

These may slightly change emphasis, but the basic meaning stays the same.

The original version is probably the best one for a learner to remember.

There are commas because the sentence contains more than one clause.

• Мне всё равно = main clause
• какой салон выбрать = subordinate clause
• лишь бы он был рядом с домом = another subordinate clause expressing the condition

So the commas help separate these parts.

A rough structure is:

• Мне всё равно, [какой салон выбрать], [лишь бы он был рядом с домом].

Russian punctuation often marks clause boundaries quite clearly, especially before conjunctions and clause-introducing expressions like лишь бы.

Yes, that is a very good translation.

A few natural English versions are:

• I don’t care which salon to choose, as long as it’s near my house.
• It doesn’t matter to me which salon to choose, as long as it’s close to home.
• Any salon is fine, as long as it’s near the house/home.

The Russian sentence sounds natural and idiomatic, and your translation captures both the lack of preference and the one important condition.