It seems reasonable, although we may still be able to send messages with radio, lasers etc. Though even that isn't easy.

Sending humans across thousands of light years seems almost impossible , but sending von Neumann probes throughout the galaxy should be possible with some reasonable improvement to our technology.

Yes. That's my assumption: a sufficiently advanced race won't send members of its own species into the inhospitable furthest reaches of space, but rather probes that can report data back to the home-world. That was always my issue with the Kardashev Scale: isn't the technological level of a species better dictated by how little energy they use to accomplish some goal?

Von Neumann probes don’t just send information back, they reproduce themselves. Anything that can manufacture a copy of itself from materials scavenged in unexplored territory can probably build anything else you want as well. A Kardashev II civilization would build a Dyson swarm around it’s own star (or one near by if they are cautious) and use a fraction of its power to send self–replicating Von Neumann probes to a few hundred or thousand nearby stars and galaxies. Those probes would build not just copies of themselves but new Dyson swarms to launch them with. Once the Dyson swarm is built there is plenty of energy available to do all kinds of things, like moving planets around, terraforming them, and seeding them with life.

There is so much we don't know. But I would happily engage in speculation:

- Without needing to make it to the closest star, we have big problems here. If we solve those problems before we leave our solar system, we may be changed beyond recognition. We may not be biological any longer, for example, or at least not forcibly so, and traveling as solid matter may seem silly to our future descendants.

- We don't understand well enough the nature of reality. For all we know, our machines and organisms made of atoms and molecules may be, by far, more inefficient and wasteful than an equivalent process at some other layer or scale. Like somebody who discovers themselves living inside a match box in a forgotten attic, we may decide to move to the more spacious main floor of the castle.

- A variation of the above: maybe space-time itself is something we use inefficiently. It could be that a way to stop being troubled by the slow speed of light is by lowering our own "life" speed, increasing our volume to span entire solar systems, and decrease our density so much that your ancestors would confuse us with sparse interstellar matter. Or, at the opposite end, it could be that we find a way to move our entire future civilization to a cubic centimeter of space and a few microseconds that feel like eons.

I don't think you need such extreme relativistic speeds to obtain the desired result though. The above assertion that every proton would have the energy of a baseball puts us at a rather incredible speed. A quick Google gives the energy of a baseball pitched at 90 mph to be around 117.4 J. For a proton to have the equivalent relativistic kinetic energy would put it at 99.999999999999999999999918% the speed of light!!

Now let's take a much more reasonable speed like 10% the speed of light. Assuming the 100 protons per cubic meter figure above, each square metre of ship now only needs to dissipate 2.27 mW of energy. 10% the speed of light is enough time to reach Alpha Centauri within a single lifetime (42 years). And fast enough to visit every part of the galaxy in less than a million years. We could even imagine generational ships travelling at 1% the speed of light (now the energy dissipation demands are 2.25 μW per square metre of ship surface). That's still under 10 million years to colonise the entire galaxy.

If intelligent life is abundant in the galaxy then I don't think the speed of spaceships at least offers a fundamentally insurmountable technical challenge for that life to spread everywhere.