How far could we effectively propel the craft?
For space travel, "how far" isn't a very good question to determine the difficulty of space travel. Instead, a much better metric is the delta-v, or change in velocity.
And to answer that question, their website says that tiny unmanned probes would be able to be accelerated to about 15-20% the speed of light.
How far could we maintain a connection for downloading information?
The current leading idea of how to manage these communications is through optical laser communications.
This will be tested over interplanetary distances for the first time on the Psyche probe, which launches in a few years; however, scientists are confident that they will be able to use these to communicate to all nearby stars with ease. We will, of course, have a better idea of what the limits are in a few years time, after we've used them over longer distances and see them in action more.