The airline allowed us to travel to our destination without checking our visas. Who is at fault for the incurred expenses?

Are Qatar Airways at fault?

Legally, no, other than perhaps for the loss of your buggy though that depends on circumstances you have not detailed.

Should all our flights be re-instated ??

No.

Are we entitled to compensation?

No.

Should they reimburse our expenses?

No.

Airlines hate it when their passengers are refused entry. It can cause significant disruption for them and may even lead to them being fined or losing their licence to fly to a country. However airlines are not responsible for the decisions of Immigration officers. The contract you had with Qatar was for carriage - and carriage is what they provided.

I do not know what the T&C's were at the time you made the contract but at present Qatar's website clearly warns:

Remember that visa requirements can change with little or no notice. It is your sole responsibility to ensure you have the correct visa and meet the local medical requirements. Qatar Airways accepts no liability for your failure to do so.

Get nasty with them and you can expect no quarter. Be nice, polite and respectful and there is a reasonable chance that you might get some concession from them (such as a discounted price for further ticket purchases from them) but this would be a gesture of goodwill on their part as they are under no obligation.

If a visa really was required Qatar will have slipped up in their procedures, so you would have a little 'leverage' but it seems what happened was that whether or not required you were refused admission (no visa perhaps just being an 'excuse').

Just to be clear:

1. You asked the wrong people for visa requirements. You should have asked the Qatar Embassy; or even queried Timatic (available at most airlines).

2. The airline check-in staff are not obligated to check your visa status. It is upto you; however as they are fined for carrying inadmissible passengers, they assumed this fine.

3. The airline was probably fined for their documentary oversight.

Now, you are not entitled to any compensation (except maybe for the buggy, but that would depend on the exact circumstances).

More importantly ... Qatar Airways did not guarantee you entry in Qatar. You are paying Qatar Airways to fly you to Qatar and that they did without issues (again, ignoring the lost buggy here).

So not sure what would be your claim?

Here is a similar situation:

1. You walk up to a taxi service counter and request a taxi to go from your home to your office.
2. The taxi service prints a voucher with your destination address, collects your funds, and then redirects you to an available car.
3. The driver asks to check your ticket stub, once confirmed the destination, drives you to your office.
4. Once you arrive at the office, you realize you forgot your employee access pass; and are thus refused entry.
5. You have to hire another taxi to take you home, to collect your employee pass.

Now, does the taxi company owe you a refund for the journey to your office? Should they reimburse you for your trip from the office back?

The situation here is exactly the same.

The OP (and wife) are the sole responsible party in the matter of not having the correct travel documents. Their mistake was calling the Home Office instead of the Embassy or Consulate nearest them. The Home Office, like my own State Department do not always have the most current or correct info on foreign countries immigration rules, especially for non-citizens like the wife.

It is not stated what the tickets were. If they were non-refundable then the OP is out the airfare, as they would have lost it anyway if refused boarding. If the tickets were refundable or date changeable, then the OP might be able to argue for compensation since if they had been denied boarding, he could have perhaps changed or cancelled them.

Qatar's responsibility is to the government for allowing the wife to travel to Doha without the proper visa. And the burden of that responsibility is to return the traveler on the next available flight at the airline's expense and potentially to pay a fine.

The OP could perhaps argue for reimbursement of the lounge fee, though it would be more of a courtesy as lounge visits aren't mandatory.

And of course the lost luggage is still covered under baggage rules, since it was checked and they did fly.

One point the OP did not touch on, but which may be guessed. Was the entire party prohibited from entering or just the wife? If the later, can we assume the rest of the family "voluntarily" decided to return with the wife. This "voluntary" choice will likely come into play in any compensation requests, since Qatar could claim the three Irish passport holders could have stayed and done the trip.