While the added crafting recipe in which a chest is placed within the recipe for the minecart works with the added wooden chests, the new chests are not compatible with the recipe in which the chest is placed above a minecart in the crafting grid.

I assume the reason the new recipe was created may have had something to do with this in some way, but this is just in the off chance that this isn't intentional.

Additionally, is it intentional that nether brick, prismarine, mushroom, and purpur chests cannot be used in this recipe?

Off topic: Is it intentional that these chests cannot be place on animals?

Possibly stupid and unworkable suggestion: Could minecarts be changed to work the same way chests in boats do (shift right click to put a chest/furnace/etc. in) so that this mod's chests could show up in minecarts? I assume this is just something where the cost/benefit ratio doesn't pan out. I don't understand programming, so while some suggestions may seem simple to me I understand they may be much more complicated or impossible to actually implement.
Now that I think about it, this couldn't be done what with minecarts with chests spawning in mineshafts and strongholds.

I think only having it work with the wooden ones is fine, for both questions.

As for the last question, very unlikely at the current moment.