The distant an interstellar object is, the more time is required for light to travel, which is known, measured as light years. So are the farthest stars and gas clusters in the Milky Way seen as how they looked millions of years ago & not how as they look today?
Most of the matter in the Milky Way is contained within a radius of 100,000 light-years from the galactic center. This encompasses the disk and most of the halo, including halo stars and globular clusters. It also includes quite a few (though not all) of the Milky Way's satellite galaxies. This is rather large - the Sun is only about 25,000 light-years from the galactic center - but not truly inter-galactic scales. Andromeda, the nearest large spiral galaxy, is 2.5 million light-years away, and so we see it as it was 2.5 million years ago. However, we see most of the objects in the Milky Way as they were no more than 100-200,000 years ago, usually much closer to the present day.
It's true that some galaxies are a couple million light-years across, but they are quite rare. In fact, the largest known spiral galaxy, Malin 1, is a mere 650,000 light-years in diameter. I should make it clear, though, that most of these galaxies are large because they have extremely large halos, consisting of gas, pockets of stars, and dark matter. Most matter is not contained in the halo. Additionally, it is not always clear where a galaxy ends and where the intergalactic medium begins.
We can't see the stars on the other side of the Milky way, but (taking some numbers from HDE's answer), If we're 25,000 light years from the center and we're looking at the far side of the milky-way, to stars at the edge of the disk, about 75,000 light years from us, we see those star systems as they were 75,000 years ago.
75,000 years is very little time compared to the 230 million years that it takes our sun to orbit the milky way, and if the outer stars orbit slightly faster (lets say 225 million years), that's just 1/3000th of an orbit, so the stars wouldn't have moved much.
What would change, if we could see it, is that stars close to each other would dance around. 75,000 years is enough time for stars to move relative to their closes neighbors. More info here and chart below.
We don't have telescopes good enough to make out details like that on the other side of the galaxy, but that's an example of something we see that would have changed in 75,000 years.
As a somewhat related sidebar, Ole Rømer used this understanding to calculate the speed of light by watching Jupiter and measuring when it's moon Io passed behind it viewed from Earth. That only works for repeating orbits though.
Yes, every object we see is how it was when the light reflected off it's surface or left it's surface(in case of self-luminous objects).
Even the sun we see is 8 light minutes old. Proxima Centauri the nearest star to our Solar System appears as it was 4 years prior, as it is 4 light years away.
Andromeda Galaxy appears to us as it was 2.5 million years ago.
If someone is present in the Andromeda and looking at Earth, then they would see how Earth was 2.5 million years ago. Theoretically they could see ancient humans roam the Earth.