I didn't fully understand the question until I watched the video. Some of the terminology may have evolved in the last 16 years or just not been common parlance for amateur bakers at the time.
What you're calling second rise is what most would call proofing. What you're calling first rise would usually be called bulk fermentation - because you often divide after, but even if not dividing by convention it's still called "bulk". In-between bulk ferment and proofing is shaping.
When bulk fermentation is finished you turn the dough out onto a (usually) lightly floured surface for shaping. You want to get rid of large gas pockets but you're not kneading. You gently stretch and fold (and roll, etc. - everyone has their own method) to develop tension so the dough keeps its shape in the oven. This would be nearly impossible to do correctly inside a bowl or fermentation vessel. Other than getting a large cutting board for this purpose to keep your counter clean - and personally I've never had great results with those vs. the smooth bare counter - there's really no way around dirtying your counter.
For proofing, the best method depends on the kind of bread. For a baguette or some round loaves you can wrap in a linen couche or just a kitchen towel to give it a little bit of structural support, and proof on a flat surface, which is what he does in the video. For batards and boules people usually use a banneton or proofing basket, sometimes lined sometimes not. Could you re-use your fermentation vessel? Possibly, if it has the right shape and you line it with a heavily floured towel to prevent sticking. But you're really better off just getting a banneton or two which are very affordable and require no cleaning, even when lined.
You can of course proof in a loaf pan and go directly to the oven to avoid the transfer step, but you're not going to get the kind of result I think you're looking for. It'll be more like sandwich bread than a crusty artisan loaf like is shown in the video.
For transferring to a Dutch oven, I find it much easier to proof in a banneton and then carefully turn it out onto a decent sized piece of flour dusted parchment. Then I lift the parchment with the loaf on it into the very hot Dutch oven. (I also tuck two ice cubes under the edges of the parchment before popping the lid on for extra steam). Works flawlessly for me. I really wouldn't worry about the temperature and the parchment. It might get a bit brittle but it's not going to combust at 450 degrees in a humid Dutch oven. After 25 minutes when you move to the uncovered portion of the bake, you can take the whole thing out, remove the parchment if you're concerned (I don't bother and still have never had a problem), and finish on a stone until it's done.
What he does in the video - lifting the dough off the flat surface and into the oven directly - is not as easy as it looks. Definitely not for a six year old.