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.