It would make Mars warmer. It would melt all the ice and CO2. It would give Mars an ocean. Of liquid rock. This is assuming that it doesn't destroy Mars completely. There might be enough fragments to make Solar System dangerous place and destroy life on Earth.

Europa is the size of our Moon. Colliding it with Mars would be similar to the collision that formed our Moon.

But would it stay on Mars? I thought the issue was there was no dynamo in the core, which lets solar winds strip it away. How long would it stay?

It would stay molten for millions of years.

Not sure if Europa's water would be flung into space, make atmosphere, or make a boiling ocean.

There's probably a way to engineer it such that the spin of the planet is 24h, the core is spun up. We'd also like ~365 day years.

At some point I saw a design for a machine you could park at the Lagrange point between Mars and the Sun that would collect solar power and spit out a magnetic field strong enough to deflect enough of the solar winds that we wouldn't need to worry about that.

Let's do it. No one remembers a coward.

Also nobody remembers when there is nobody alive

And then we completely skew Mars's orbit until it crashes into the sun or is flung out of the solar system.

If we had the technology allowing us to move a full satellite through the solar system, we could probably do it in a way that would just make Mars a bit closer to the sun so that the weather gets nicer (sure, if it gets too close to earth it's going to mess up with both orbits, but we can as well correct it when it happens, right?)

It doesn't have to be near the sun to have heat or be warmed by the Sun. It is still currently in the goldilocks zone, same with Venus. The difference is how well each planet traps heat. No atmosphere no heat for mars, highest peak surface temperature is 70 fehrenheit or 20 celsius. Not much but enough green house gas and you could raise it by 20 degrees or more. Other thing to consider is that you don't need to move Mars, you could create artificial magnetospheres.

Might have some "slight" orbital and shrapnel repercussions..

If we could crash Europa into Mars we'd have capability to terraform Mars in a more reasonable manner.

And then wait 160 million years for the planet to settle. I like your long term thinking.