No. Checkered giants don't show up in the pet trade. They are very large and usually extremely aggressive. A very rare breed. Any breed can have that pattern. Its the long, full arch body type which distinguishes english spots and checkered giants, which these rabbits just don't have.
Images are showing up now. They are mixed breeds, that is as far as we can tell. Definitely not english spots. They don't look anything like english spots. Most breeds come in that spotted color ("broken pattern") so who knows what its ancestry is.
Mother is a Sable Agouti (aka sable chinchilla or shagouti). Father is a Black Tortoiseshell (aka tort or black tort). Babies are (in this order) Orange, Chestnut, Black Otter, Cinnamon (aka chocolate agouti and in rex furred rabbits as amber), Fox (aka torted otter)
It's not possible unless there is the steel gene involved. A steel rabbit that carries non-extension/orange can sometimes appear solid black, despite being agouti, which can lead to confusion. Steel is not particularly common netherland dwarfs, though - unless this rabbit had steel in its...
Looks good for fawn to me
Mini rex usually have bright orange pigment in their coats, so she looks dilute. Dilute mini rex isn't a standardized color so it is known by a few names, usually fawn or cream
Mostly likely chestnuts (or if this is a rex furred breed, castor). You'll likely get some black otters as well, as the fawn carries self which is recessive to otter. All of them will carry non-extension and probably tan pattern as well, which is not a combination that people like to see in show...
Darker one looks like a sable point. The himalayans are 2 black and 1 blue; however with a sable point sibling I would not be surprised if they ended up being non-extension himalayans, but that would be very hard to tell at this age.
"solids" are any rabbit that isn't broken. The term for a rabbit with one single color like black, blue, chocolate or lilac is "self"
To get oranges or black you'd probably have to bring in different colors. Black is dominant to harlequin. For oranges you'd need to bring in some non extension...