Can my wife being Indian National, go out of transit area for some hours without a visa in Schengen Airport?

KLM Visa Information, back-ended by Timatic Web, which in turn is used by most airlines, says that your wife, travelling to Mexico from India can Transit without a visa in Belgium, provided:

• she has onward ticket,
• stays international/air side,
• has visa for destination,
• and same calendar day (?)..

1: Is possible for my wife to remain in the Brussels airport transit area for the 16 hours?

Yes, she could stay airside, but you will know this already in Delhi before boarding, because the airline will make sure your wife (& you & all passengers) has all the required visa conditions/necessities for all transit & destination points, otherwise they are going to get massive fines(if they haul people to ports & then if those are not allowed to enter/proceed by country). At the same time, as these are separately purchased tickets (from different airlines?), as far as first airline is concerned, Brussels is your destination for them.

Although some of the front-desk employees might not know about the exceptions, so its always good to speak with a supervisor, & keep some printouts from Timatic System. As your tickets are separately purchased, it may get some unwanted attention from the check-in staff.

I don't need a visa, and would collect the baggage for both of us, wait in the airport, check in for both of us the next day.

Now, as you don't need EU Visa, you could always enter Schengen (pass through, and check-in to next flight), but for the wife, best place to ask if she can "check-in" from airside is the airline itself. When you do, ask them also if you could give them your wife's luggage. (or they might want to label all luggage as your's (one passenger) for security & economical reasons), & thus might want you to pay for all the now excess weight.

2: What would be the consequences of my wife arriving in Brussels without the necessary visa?

Like I said above, there is very little chance that airline will allow your wife to board if they thinks she does not have necessary documentation (for transit & destination), & if she is not eligible for TWOV.

Can we explain the situation to the immigration officer, and be allowed to remain in the airport for the 16 hours if we can prove an onward flight the next day?

You will encounter Immigration only if you attempt to come to land-side of Belgium. Once you are past the Immigration Line/boundary, there is no more boundaries separating airport area from rest of the country.

3: Is there a JetairFly counter inside the transit area?

Best way to know is to ask JetAir..

If my wife doesn't need a visa and she remains airside, can she check in for our flight to México?

Only if airline has check-in counter in air-side.

I have attached screenshots from mobile, I will improve this answer and formatting from pc, as as of now our fibre connection was damaged.. Edit: I did, but do not know how to strike the text. Edit: found how to strike, use <del> ... </del>

Screenshots from KLM Visa Info Timatic Page:

One can't stay in the Transit Area and collect the baggage. To collect the baggage she would have to cross the EU border first.

If both legs of her flights are flew by airlines which have baggage agreement with each other, she can probably send baggage to your destination directly (México) but she'll have to wait the whole stopover in the airport's Transit Area. Making her two flights connected. Ask your airline explicitly.

I'm not aware of EU-issued transit visas on the spot.