I'm interested about changes in the core and mantle behavior, a
potential magnetosphere, changes in the atmosphere and tectonic
A lot of this is very theoretical. Venus' atmosphere is nearly 100 times as massive Earth's atmosphere. It's also nearly all CO2, a greenhouse gas, which means it's good at trapping heat. That wouldn't change if Venus' rotation rate was faster.
"Core and mantle behavior" is a bit of an unusual statement, cores rotate with the planet and they slowly cool, there's some convection vs conduction going on, but it's difficult to study the cores of other planets. It's hard enough studying the core of Earth. But what we know of the core of Earth, or think we know is that Earth's core and Earth's crust rotate at slightly different rates. This may be due to Earth's crust slowing down over time and the core, not slowing down as fast. There's a slight variation in rotation rate. Also, Earth's inner core is solid, Earth's outer core is somewhat liquid (though extremely viscus), and as it cools, the inner core slowly grows, the outer core shrinks and in the process heat is released. This process creates a flow, which may be partially influenced by the Moons orbit, and together the circulations are thought to create Earth's magnetic field. This isn't certain, but the flow of the outer core is thought to be the cause and Earth's rotation may play a role in that.
So if Venus rotated much faster, in combination with it's proximity to the Sun and the solar tidal forces, it's possible a faster rotating Venus would have a permanent magnetic field. It's far from certain, but that's a possible outcome.
Plate tectonics is more tricky. Various studies suggest that planetary size has a lot to do with plate tectonics, and abundant water plays a role too, oceans making plate tectonics easier. Venus is on the small size and it has almost no water, so it's not a good candidate for plate tectonics. Venus also has very little granite. This is geology more than astronomy, but abundant granite (IMHO) plays a role in plate tectonics. The rocky bodies in our solar-system. Mercury, Venus, Mars, the Moon, are all, basically basalt like on their surfaces. Earth formed large granite land masses that float above the more dense basalt like layers under the oceans. Without granite land masses and without oceans, I don't think Venus is a good candidate for plate tectonics.
Did Venus ever have plate tectonics, like billions of years ago? Maybe. I have no idea, but maybe. Would it have plate tectonics if it rotated faster? I doubt it. A magnetic field - maybe. Plate tectonics, I don't think so.
This question is probably too general for stack exchange and it may be closed, but I don't believe that just spinning Venus faster would change all that much apart from maybe giving it a magnetic field, and that wouldn't change all that much on Venus. It has an induced magnetic field anyway.
If you could give Venus an ocean, maybe make it more massive and move it further away from the sun, then some interesting things might happen like plate tectonics, which could have lead the planet down a different path, perhaps more similar to Earth.
Earth was able to absorb CO2 into it's vast oceans and take in CO2 by photosynthesis and rocks were able to absorb CO2 by plate tectonics, and Earth was far enough from the sun to not have it's surface water evaporated and lost like what probably happened to Venus.
I think planetary formation, based on size, type of star, temperature, initial materials, etc, will be a very interesting field of study when/(if) big enough telescopes are set up to actually get a look at distant planets. It's very difficult to even observe, much less, study a planet that's light-years away. The so far twice or three ties delayed James Webb Space Telescope might get the ball rolling on this subject with better observations of exo-planet atmospheres, but it's a long ways away from being well understood. The James Webb ST is an enormously difficult undertaking, so we should be patient. With luck, it'll be up and taking in images within a few years.
Finally, would Venus' rotation, if it was faster, affect other planets? Basically not at all. Jupiter's rotation doesn't affect Earth. Rotation rate doesn't affect other objects at great distances.