How Product Hunt really works

This is how I think Shark Tank and Dragon's Den work too. The "sharks" all parade their new products on TV under the guise of entertainment, and throw in a couple silly/heartwarming ideas and people.

You can find a lot of testimonials from people who have appeared on Shark Tank, and all the ones I've seen say "I was surprised by how real the show is". The only major change is that there's an hour-long pitch/discussion that gets edited down to 5 or 10 minutes of interesting TV. Nothing is scripted, the sharks don't know what's coming, the negotiation is real, etc.

I guess it's possible that these people were paid to lie about the show, but it seems like Shark Tank is one of the most realistic reality shows.

The ideas aren't mutually exclusive; There can be real people and planted advertisements...

Incorrect, at least with Shark Tank. Can't speak to Dragon's Den. I worked for one of the Sharks on their ST accounts. Nothing is staged, it's all regular people who submit their ideas and are selected based on who the network thinks will make for good entertainment (personality, "wow factor" with the product, etc). During filming, the sharks are given an overview of who the person presenting is and what their product is/does. Everything else is organic. The Sharks have no advance knowledge of the products and the deals get made or shut down after due diligence is done after filming.

That's not to say that the sharks don't do a huge dog and pony show of promotion around their products, though.

What type of due diligence must a shark or their team do during the show? I imagine these funding offers are just term sheets pending a legit financial review?

The due diligence comes after filming. The deals you see on TV are a handshake deal and in my estimation only about 30-40% of them actually end up going through because a lot of founders lie about things like being litigated against, having tax liens on their company, etc. The sharks all have people on staff who sniff these problems out and make sure everything the founders say about their numbers are true.