Yes, the Birdy lock up in one piece when folded. The way the front wheel folds is totally different of the Brompton, so people used to fold the Brompton need to learn how to quickly fold the front wheel of the Birdy, once done its very easy o do, the folded front wheel is locked by the folded stem (the rear is very similar, folding on the rear suspension axle, rear locked by the seatpost pushed down).

For the shopping cart mode, its more or less like the Brompton: a naked bike without any mudguard or rear rack doesn't roll (like the former E Brompton that doesn't really roll folded).

You need to add some rack or/and easy wheels to make it roll folded.

R&M rack roll folded but not like the former R Brompton that rolls on 4 easy wheels, it has only two easy wheels to pull it.

There are third party rack with 4 easy wheels.

For the folded size, the bike is the same, its just that the saddle is mounted differently by R&M = for tall people and that the R&M Birdy come with mudguards from factory. Just like the Brompton, where the R (mudguard+rear rack) is bigger (higher) folded than the E, the Birdy with rear mudguard is also higher folded.

Its very easy to remove the seatpost+saddle (its a quick release lever like the extension of the Brompton telescopic seatpost, no tools needed) from the Birdy to make it smaller folded and the Birdy seatpost is long enough to fit for tall people, there is no optional longer and telescopic seatpost that both make the folded Brompton higher folded.

I have the BW case, its too big for airlines rules and its too small for a Brompton with 600mm seatpost and saddle on the top position of the pentaclip.

Now, I never said that the folded Birdy is as small as the Brompton, but very close, so close that the slightly bigger fold isn't a problem in daily life and the BIrdy is much faster and comfortable than the Brompton. Its also more versatile for luggage transport with now the possibility to use a Brompton front bag, front low rider pannier and rear full size rear pannier (even for people with big feet).