These words are all very close in meaning, so I'm not surprised that people have trouble distinguishing between them.

Benefit is the most frequent and the most general of these words. It can be both countable and uncountable, and it means an advantage that you get from a situation. It is used in many different patterns and constructions (take a look in the Macmillan English Dictionary (MED) to see the full range of these):

Consider the potential benefits of the deal.
The new sports facility will bring lasting benefit to the community.
Not all competition is of benefit to the consumer.
He has had the benefit of the best education money can buy.

Gain is very similar in meaning to benefit: a gain is a benefit or an advantage. Gain is countable and is neutral in register:

It is a policy that will bring significant gains to all sections of the community.
Surely there could be no gain to him from the old lady's death.

There may be a suggestion with gain that you have made an effort to achieve something, whereas a benefit can be undeserved. This is shown in the expression 'no pain, no gain', which means that you have to suffer in order to achieve things.

Profit is defined in MED as 'the advantage you get from a situation'. It is uncountable (so you don't talk about profits, except in the financial sense) and is labelled as formal. It is much less frequent than the other two words, and is rarely used in speech:

No profit lay in that line of thinking.

Should we say:
Cheese and milk are / is in the fridge.
Every boy and girl is / are clever.
Here come / comes Mr and Mrs Tan.

Strictly speaking, it should be: Cheese and milk are in the fridge and Here come Mr and Mrs Tan because the subjects are plural. However, people often use a singular verb with a plural subject in speech, so it is quite possible to say Cheese and milk is in the fridge and Here comes Mr and Mrs Tan and it would not sound wrong to many people, though some might object (even though they might in fact say such things themselves without realising).

I wouldn't say either Every boy and girl is clever or Every boy and girl are clever because both sound completely unnatural. But with a different verb – Every boy and girl wants to be popular – it becomes clear that the verb should be in the singular, even though the subject is plural. This is just an oddity of the language. If you look at the entry for every in MED it says:

A noun subject that follows every is used with a singular verb.

Each behaves in a very similar way. One way of remembering this is to think of the famous lines from Gilbert and Sullivan's operetta Iolanthe:

That every boy and every gal
That's born into the world alive
Is either a little Liberal
Or else a little Conservative.

Could you possibly help me with the following expression found in an activity: … you don't get one thrown in with the room, don't worry. Be happy. I would appreciate if you could tell me what get one thrown in with the room means and if it is a fixed expression, collocation or idiomatic expression.

It would be useful to know what one refers to, but the meaning of the expression is clear without it. If something is thrown in, it is included in something you are paying for at no extra cost. Here is the example from the Macmillan English Dictionary:

Buy a computer now and get a free printer thrown in!

This means that the shop is so eager to sell you a computer that it will give you a printer at no extra cost, in order to encourage you to go there rather than to another shop. (Whether you can ever really get anything for nothing is a different question, and not a linguistic one!)

And here's another about children's parties:

Swimming, football, party games, bouncy castling or rollerblading to disco music – all are on offer at the Oasis Leisure Centre in Swindon. A meal, a toy and a fizzy drink is thrown in with every booking.

In this case you pay for the activity and the food, toy and drink are free.

MED classifies this as a phrasal verb, because it consists of a verb (throw) plus a particle (in) and has a meaning beyond that of its constituent parts (that is, it doesn't mean the same as throw in in the sentence 'I threw a stone in the water'.) To this extent it is idiomatic, like all phrasal verbs.