Where can we store our luggage after checking out so we don't have to book an extra hotel night?

You certainly have more than one option, when it comes to leaving your stuff so you can move around freely:

• You'd definitely want to ask to the personnel of your accomodation (hotel, hostel, apartment, etc.) if they can store your luggage until you are ready to leave. Most of the accomodations offer this kind of service and usually have a locked room, where all the stored luggages are collected. I was never asked for a fee when I used this service.

• If you are very close to the airport, you could store your luggage there. I don't know any airport that has no baggage-store. This is likely the most expensive option.

• If you are near a train or underground station, you could look for self-storage areas, where for a couple of Euros you can lock your small trolleys or backpacks inside secure lockers.

I don't know any specific Info about France or Euro 2016, but what I wrote is pretty much international.

Most hotels and many hostels will have a space you can leave your luggage and collect it later. Enquire about this at the front desk. Some hotels even offer this service to non-residents (although usually for a fee in that case).

In the event that you can't make use of this, for example the hotel doesn't offer the serivce, you are staying in self-catering accomodation without a concierge, or the hotel is simply inconveiently located, then you have various options. Primarily what you are looking for is a "left luggage" or "luggage locker" service. These can often be found in or near to train and bus stations, as well as airports. There will be an assosciated fee, which can vary from trivial to fairly steep, depending on location. Another option is to enqure if your airline will let you check in your luggage early - some will take luggage up to 24 hours ahead of time. Depending on the airport location, the practicality of this may vary.

You can browse the luggage-storage tag on Travel Stack Exchange to see where questions have already been asked about the availablity of left-luggage and locker facilities.

1. Basically any hotel will keep the luggage for you - secured - until you pick it up. That might be inconvenient if the hotel is on the other side of town, though.
2. You can check your luggage in right away. Often, airlines allow luggage check-in up to 24 hours ahead of time, but you should verify before you drag it to the airport; and again, that could be inconvenient.
3. Other local storage depends of course on the city you are in... there can be no general answer.

You don't mention what country, but I am assuming this is Europe and not USA as it would be very odd socially to hang out in an airport that long here and could even get you in trouble with TSA if someone decided to notify them. Nonetheless here are some options you can use (at least in the US):

• Early or Late checkout. In your example you're leaving a bit too early to really use this, but many hotels quietly offer a early checkout option. Leave early enough and you pay less than if you stay the full day. You could stay until 8pm the day of the flight, and use this service to only pay for a portion of the cost of a full day. This is far more common at hotels near airports or resorts, and is dependent on the hotel itself.
• Leave it in the rental car. Rental companies are pretty flexible about return times. So you can leave the luggage there. For most there is no difference in cost in returning the car at 6am or 8pm. This is not true for all rental companies though.
• You can carry it with you. Get luggage that you can carry around easily. This depends a lot on how much you have to carry, but it's not unusual to see restaurants around airports with people in them with some carry on luggage.
• My all time favorite is to UPS the luggage home. Depending on how much you have it may be more expensive then staying at the hotel. But it is a good way to get your luggage home.
• Look for other ways to save money. Just gotta throw this one in there. At least here, getting to the airport that early would be a nightmare. Spending 6 to 12 hours in a US airport would likely drive me to tears. Renting a normal room for another night would would be around $40 at a cheap hotel. I would strongly recommend using the grocery store instead of a restaurant for a day or two and spending the $40, over trying to do anything in an airport for that long of a time. That said, US airports (and flying in the US in general) has really become quite bleak. There are no restaurants, no stores, no entertainment, nothing but chairs once your out on the concourse. Sitting still for 6 to 12 hours with bad WiFi, and not even a place to get a drink would be, well, ill-advised. Keep in mind that every airport is different and you may like the quiet time, etc etc.

So I would say, talk to the hotel, see if you can do an early checkout. Even if you can't, just try to save that money elsewhere on the trip. Whatever you do, try not to sleep in the airport. You may be left alone, you may be hassled, you may be arrested (specially if you're at the public part of the airport).

There is a luggage storage service in Denver called LUGDEN. They are right by Union Station and offer luggage storage by the hour, day and week. Their website is lugden.com