Mankind's yearly energy consumption early 21st century was around $5\times10^{20}J$
Each year, while the sun outputs about $1.2×10^{34}J$ per year, which means we use about 10 femto-suns of power. To Nikolai Kardashev reaching out to the sun seemed like an obvious thing to do. Think about it: you have this immense amount of free energy streaming out into space, and essentially every Joule of it is wasted. You don't have to build a nuclear reactor, you don't have to worry about fuel. All you have to do is reach out and harness it. Since the intensity of the irradiance descreases with the square of the distance, you can maximize your capture and minimize the surface area needed for a particular amount of power generation by placing your generators closer to the source.
Solar Irradiance at the Planets
Planet Solar Irradiance, W/m-2
Mean Perihelion Aphelion
Mercury 9116.4 14447.5 6271.1
Venus 2611.0 2646.4 2575.7
Earth 1366.1 1412.5 1321.7
There is a vast, vast amount of matter in the Solar system, some of it conveniently outside of the massive gravity wells of the rocky and giant planets, so it's not an insane guess to expect that we'd use Asteroid Belt matter first, then the even larger Oort cloud resources. Obviously we wouldn't just make a Great Leap Forward and become a Type II overnight. We'd first have to reach Type I, defined roughly as making use of the resources of a home planet. If we take the Yearly solar irradiance of Earth, at $5.5×10^{24}J$, we still have a ten-thousand fold growth curve to ascend to even reach Type I. To do a Fermi simplification, let's assume 100% capture efficiency, so if you built solar panels at the Mercury perihelion, to reach Type I via solar you'd need 12 million sq. km. of panels, which is in the same order of magnitude as the area of Europe. Might seem like a lot, and it would doubtlessly require far more resources than we can currently even dream about harnessing, but the area of a sphere at the orbit of Mercury's perihelion is about 6.6e15 sq. km, so you've only built about 2 billionths of a Dyson sphere. Yeah, space is BIG. You can see from that that you can go a long way towards a Dyson sphere before anything at all would be noticeable on Earth, and with some level of planning, you can ensure that even a near-complete Dyson sphere does not shade Earth (or the other planets) at all.
We got a long way to go.
Ok, let's talk mass requirements, at a conservative 840 tons / sq. km, the structure required to bring us up to Kardashev I would weigh about 1E13 kg, less than the mass of Mars' Phobos, easily achievable using Asteroid belt materials (ignoring ancillary structures for energy storage, transmission, repair, etc, -- you can quadruple my estimate if you want, and then triple it again if you want to assume 30% efficiency, which still leaves you within an order of magnitude of the first estimate anyway). For a full Dyson sphere you'd need at least 5.5E21 kg of mass, which puts you around the combined mass of the asteroid belt. Throw in a few hundred massive Oort objects if you need to. So doable without dismantling planets. We might need to to some transmutation of materials, but with so much free power, shouldn't be a major issue.
Now, to the question of whether a genuine Type II civ would actually build a Dyson sphere, we can't really know. Perhaps a civilization so advanced has found far less crude methods to extract energy than from the wiggle of electrons on a slab facing a natural fusion reactor, from burning complex carbon molecules in a tin can, or from using atomic decay to boil water and using the vapor to make some brushes spin.
I recall reading once that there is enough zero-point energy is the volume contained by a regular mug to boil all of Earth's oceans away. And that's the stuff we know about. Who knows what wondrous tricks the descendants of Humankinds will come up with in the future?