1) How can I check that all three access points (the two airports and my router wifi) are in the same wifi network?
If you configured the AirPorts in Bridge Mode, they are on the same network as the Thompson router. You can verify this if you can check the client device list for the Thompson router. You will see that the AirPorts are on the same network.
Everything will behave as if it is on the same network if you used the same wireless network name, same password and same wireless security on all the wireless routers.
the two airports and the router are setup exactly the same. including the channel.
Unless this means that you are using the "Automatic" channel setting, you want each wireless router separated by at least 3-4 channels from the others. So, if the Thompson is on Channel 1, then manually set one AirPort at Channel 6, and the other at Channel 11 or similar.
As far as 5 GHz, just set the routers to use a different channel....you don't need to worry about separating them by 3-4 channels on this band.
Looking at the airport utility, it seems that the wifi clients on the 2nd floor are connected to the airport in the 1st floor instead of the one on the 2nd floor (which is closer) and --as expected-- they don't get an excellent signal.
This will depend on what "clients" you are talking about. If you have a recent Mac laptop, you can "walk" it around the house and it will automatically switch to the wireless access point with the best signal.
If your "clients" are iOS devices like an iPhone or iPad, this will not usually occur since the iOS operating system is greaty simplified.
iOS devices tend to "hang on" to the access point with which they originally connected. Even if you move the iOS device closer to another AirPort, it may not "switch" to that AirPort. Nature of the beast with iOS.
If you move the iOS device to another area, you will likely need to either power it down and power back up, or turn off the wireless on the device and turn it back on so that the iOS device will hopefully switch access points and pick up the stronger signal.