Very little. They basically wanted to do a TV show about Mongols but needed a white main character to appeal to a western audience. Not much specific is known about what he did in China so the show writers are making it up as they go along, using him to try make an interesting story even if there was no historical events to back it up (and often they're completely making things up/misportraying historical events like Kublai's brother rebelling against him, or Kaidu calling another election, all nonsense.)

Did he REALLY spend time in Kublai Khan's royal court?

Yes. Marco, his father and his uncle delivered holy oil from the Pope to Kublai Khan. They stayed in the Khan's service for 17 years. Marco claims to have served as diplomat/official/tax inspector and accompanied the Mongols into battle. How much of this is true nobody can really say, some accuse him of exaggerating and aggrandizing himself in his book. Not to mention the liberties the writers are taking with his story.. No he didn't reinvent the counterweight trebuchet to save the Mongols from a failed siege, no he didn't bed various Mongol princesses or save the Khan's life from assassins etc etc.

Was he REALLY left there by his merchant father?

No. His father and uncle stayed there with him, they didn't leave him behind as a bargaining chip, they weren't exiled and branded by their son/nephew. Show writers wanted some family strife and so made it up I suppose. I'm apprehensive of what else the writers are going to make up. So far I'm at the point where he's meeting/plotting with the Pope and Mongol pedophile priest. I'm not hopeful.

If you want to know about Marco Polo and his travels, what better way than to read his journal/autobiography, The Travels of Marco Polo? It's a fairly thin book and directly relevant to your interests.