There's an excellent 2-page spread about these guys in the Last Jedi Visual Dictionary. They are called Praetorian Guards, and they are mysterious by design. They wear full-body armor and masks that hide their identities, and are intentionally supposed to be "anonymous".
Here's an excerpt from the Visual Dictionary:
The Praetorian Guard are a bold example of an Imperial symbol
reimagined, distorted, and aggrandized by the First Order. These elite
sentinels stand watch over Supreme Leader Snoke's throne room aboard
his flagship. Their brilliant red uniforms are a deliberate echo of
those worn by Emperor Palpatine's Royal Guard, but the pageantry of
the robes has been swept aside to allow an unhindered view of the
precision-machined combat armor. The Praetorians are the ultimate
close-circle guard, eschewing ranged weaponry. They are the last line
of defense protecting Snoke, ready to destroy any threat that would
dare penetrate so deeply into the First Order's heart.
I can't perfectly answer the last question (why they were still fighting when their master was dead), but evidence seems to suggest that it was out of pure loyalty. Since Snoke can clearly read minds, he probably chose those 8 guards specifically because their loyalty was unquestionable.
There is also a section about their uniforms that specifically references their loyalty (some slight paraphrasing to cut down on the extra details):
The layered armor of the Praetorians is... impregnated with conductive
wirepaths that, once powered, create an intense local magnetic field.
Once activated, the powered plates can deflect blaster fire. Even a
lightsaber will glance off... The mag-field exposure is ultimately
painful to the wearer, but... the Praetorians endure this out of
unswerving loyalty and duty.