As an opening aside, what do you mean 'duvet-style' brigandine? I am not acquainted with that term. Do you mean the kind of later 16th century jack of plates that includes a cloth exterior and visible quilting? This is a somewhat different piece of armour from the 15th century brigandine like the one that you illustrated (though the basic principle of plates attached to a cloth garment remains the same).

Because my research into brigandines focuses on the 15th century and because you included that one in your illustration, I will focus on 15th century brigandines.

Breaking up your questions:

• Was supplemental armour worn with bridandines? Yes. The illustration that I used for a 15th century brigandine above shows short mail sleeves worn with a brigandine, as well as a mail collar of some kind. This could be a mail shirt, but based on inventories of the 15th century it may well be supplemental pieces of mail worn independently. In the secon half of the 15th century the household accounts of John de Vere, Earl of Oxford list 101 brigandines, 77 pairs of splints (cheap plate arm defenses), 84 mail gussets (defenses for the underarm area), and 24 aprons of mail (skirts). They do not list mail shirts - all the mail armour is supplemental. Mail standards/collars would have been particularly important for defending the lower neck, and you see them worn underneath brigandines in many illustrations. In addition to supplemental mail and plate defenses, brigandines sometimes had spaulders constructed in a manner similar to the rest of the brigandine as in this 15th century portugese tapestry.

• What was worn under brigandines? 'Gambeson' is a term from earlier in the middle ages referring to a tunic-like padded or layered form of cloth armour. They wouldn't have been worn with brigandines. The garment that you are describing seems to be an arming doublet, a foundational garment worn under armour that armour could be laced to, including supplemental pieces of mail. I am not aware of the use of arming doublets outside of the context of a full harness of armour by men at arms. That does not mean it was not use, but I have not seen it depicted (and our literary sources for common soldiers are scarce). In the ordinances of Charles the Bold there is a requirement that archeers be armed with a brigandine and a 'padded jacket'(this is the translation that appears in Strickland and Hardy's Great Warbow). The jack a 15th century form of fabric armour, so this is one case where we can see fabric armour being used to supplement brigandines.

• How was supplemental armour attached? This is hard to say. Some illustrations, such this 15th century portugese tapestry show the spaulders of the brigandine pointed (tied) onto something, possibly an undergarment. Other equally detailed illustrations do not, such as this painting, 'The Resurrection Triptych' by Memling. Splints, plate arm defenses, would need to be pointed onto something at the shoulder at least to hold them up (surviving examples from the 16th century have lacing holes at the top of the shoulder). All of this would require some form of garment under the brigandine to tie the points to, but it is not clear to me what this was. Finally illustrations of mail with brigandines seem to show the mail going -under- the brigandine and show no obvious means of attachment. In general supplemental pieces of mail are attached separately in medieval European armour (excepting aventails on 14th century helmets). This all suggests that it was attached to some underlying garment or held up on its own (as with a standard for the neck, or a skirt) rather than attached directly to the brigandine.

Sources: Images as cited

Strickland and Hardy - the Great Warbow - details on the ordinances of Charles the Bold

Edge and Paddock - Arms and Armour of the Medieval Knight - de Vere inventory

Thom Richardson - The medieval inventories of the Tower armouries 1320–1410 - though this thesis focuses on the 14th century, it deals with the shift between mail shirts and smaller supplemental pieces of mail in England.