Connecting Freebox to new DTI box?

Communication boxes are independent of ISPs and have nothing to do with them.
It is certain that the manufacturers of communication boxes could make the effort to produce an explanatory booklet for users because it is not obvious at first glance.
Again, here you have a manual box; there are much more complex automatic boxes.
It is true that the test socket should not be used in principle, as it is reserved for line testing, its use isolates the rest of the installation for the telephone, but since you will not use it for the telephone, as almost no one does today, it doesn’t matter.
Similarly, you should have a master filter, but it is only useful in partial unbundling, which no longer exists in new installations.
Indeed, the dangling cords are intended for telephone distribution on wall sockets from the DTI, without passing through the test socket; this is a telephone micro-distributor also called octopus cable.
You could also connect your ADSL from your box to one of these 4 sockets, but you will need to buy a female RJ45 to male RJ11 adapter or a coupler to put a male RJ11 cable at the end.
Indeed, you will need to add 4 patch cords of about 50 cm to distribute the ethernet ports of your box to 4 sockets as well as 4 cords to connect the wall sockets to the devices, 5 m recommended.
These patch cords are sometimes provided by the manufacturers of the boxes, but electricians rarely leave them.
Similarly,
the electrician should have identified the sockets to determine their destination; you will have to do this yourself, I recommend labeling them.