Flooring – How to trim between wood floor and circular opening for spiral stairs

You can but the if its an exterior facing wall you'll likely cut the vapor barrier if you don't use the proper tool. I recommend the Fein Multimaster for this. It has a depth stop gauge attachment so you cut just enough of the drywall without penetrating all the way through. The segment saw blade is best for this. A ton of companies are making knock offs of this tool that you can find in big box stores now so they may work too. Fein is going to be more expensive but it lasts decades.
Another option to consider is to just pull out and replace all the baseboards with something thicker and looks better than what you have now. If you choose this option heed my advice and pre-prime, if not already, and pre-paint all the boards with no less than two coats before installation. You'll still have to apply a final coat after installation to cover nail holes, but you can go extremely fast because you've already pre-painted.

Not only have I installed a few tavern grade floors but I bought both from build-direct.

I think some people are confused. This is not engineered. You will have a LOT of variety. So you better like all of the character or you will throw literally half out and any savings will be out the window.

As far as holding up to kids and dogs I would do a thorough scratch test. Really you are testing the top coat. Take a hammer to it, a screwdriver, and put on some heels and dig them in. Whenever doing a floor from build-direct we get at least 5 samples and go over the durability of each before purchase. I am very surprised at the the variance, especially with the engineered.

For the wood you picked out, it requires nail down. Which means you will need to put down a subfloor. There are hardwoods that you can glue down to concrete subfloor but most "home" subfloors are not smoothed enough to support this type of install - which is mainly for apartment buildings. You would need to take pictures of your subfloor to know for sure. Anyway I certainly wouldn't plan on doing an "odd" install with tavern/utility grade.

I am not sure what type of dog you have but I am guessing you have one that would scratch a wood floor. So I would suggest that you take the engineered flooring out of the equation. Which means you can float vinyl planks - there are some that look a lot like the Jasper you want or you need to get the subfloor put in.