It is true that distant galaxies within clusters can move thousands of kilometers per second, and that the expansion of the universe constantly 'dilates' the distances between clusters of galaxies. When it comes to nearby stars, we can measure both their velocity components away from us via the Doppler Shift, and measure the two velocity components across our line of sight via their annual proper motion across the sky. We can, then, know with perhaps 10 percent accuracy just how fast a star is moving, and predict its orbit for a few thousand years. For galaxies, we can only measure the redshift of the galaxy, which for nearby galaxies is essentially their Doppler Shifts, but for more distant ones, becomes a combination of both Doppler and 'cosmological' redshift. We cannot, however, ever measure the speed of a galaxy across our line of sight because in 100 years even the Andromeda galaxy will only move about 0.032 light years, and more distant galaxies even less. Over a period of 1 billion years, a galaxy moving across our line of sight at 1000 kilometers per second will travel 1 million parsecs so relative to a typical galaxian size, this is a significant distance.
How do astronomers deal with such a '1-dimensional' universe? We rely heavily on statistics. If you measure the radial velocities of a dozen galaxies in a cluster, their speeds form a distribution 'function', and it is a simple matter to infer from this distribution function what the most likely velocity components of a particular galaxy are given its location in the cluster. This 'kinematic' analysis can be strengthened substantially by realizing that the speeds of galaxies are not drawn randomly out of a hat, but are dictated by Newton's law of gravity. So, you can infer what the components of a galaxy's velocity are across the line of sight even though you never see them move!
Now, the question asks what implications this statistical 'indeterminism' has to our confidence that we really know what the universe looks like. Well...define 'Real'. Whether we know it or not, our brains constantly work to extract the best statistical information out of the physical world that it can in order to find rules that help it survive. Sometimes this leads to the brain seeing patterns of cause and effect that are not there such as astrology and other 'rules of thumb'. Of course we will never really know exactly where a distant galaxy is RIGHT NOW, with a precision of 1 light year or less. This isn't very important because it is only where we see it now that is important. The rest is simply extrapolation using the regularities built into the way physical laws seem to work.
I am reminded of the joke about the person that jumped out of the 34th floor. Passing the window on the 13th he was heard to say 'So far, so good!'. For some systems in nature driven by blind gravity and a couple of conservation laws, all you need to know is how they started out, the rest and the destiny follow automatically. So long as cosmologists get the overall picture correct in a statistical sense, the details and the 'reality' follow.