Does the Universe have weather at full scale?

If everything is moving and the distribution of matter in the Universe isn’t evenly spread, are there whirlpools, eddies, gusts and such, of the galaxies, clusters and whatnot?

Yes.

A lot of galaxies are whirlpools, eddies, gusts and such.

The universe is expanding such that some parts have no possibility of ever interacting with some other parts - this doesn’t mean the universe doesn’t have weather, but it does impose a limit on the scale.

I don’t think there is enough interstellar air for the formation of hurricanes or tornadoes as we know them. Arguably, the shape of a galaxy itself resembles a storm and there may be some physical principles that apply to both. Galaxies can combine with other galaxies and objects entering a galaxy might end up in orbit of the center, making an analogy with getting “swept up into a storm”.

But there are, I presume, high pressure regions and low pressure regions that you can treat as very, very compressible fluid.

What weather patterns really don’t have is gravity. No high pressure zone on Earth ever gets so dense that it will start to suck in surrounding air through gravitation in any significant way – inertial forces are orders of magnitude larger than typical air-on-air gravitational effects in the atmospehere. But trying to model most astrophysical phenomena without taking gravity into account would be pointless.

Well it does take some stretching of the term ‘weather’ when describing the universe. Spiral galaxies resemble hurricanes, but the ones I know of aren’t spinning very fast, and they’re not being fed by warm ocean water.

If we want to compare the galaxy to weather, I don’t think a hurricane is the right comparison. As I understand it, the spiral arms of a galaxy are actually compression waves moving through the galaxy - and moving at a different speed from the stars so that stars pass into and out of the spiral arms. So there’s something of a high/low pressure differential there and it’s enough to affect star formation.

On the scale that we usually call “weather”, the gas particles are molecules, and the interactions between those particles are all ultimately electromagnetic. On galactic or cosmological scales, we’re still dealing with gases, but the particles are stars, nebulae, clusters, or galaxies, and the interactions between those particles are all gravitational. This makes surprisingly little difference for purposes of analyzing these systems.